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= CD DO 1 66492 3 m FUNCTIONS OF A COMPLEX VARIABLE WITH APPLICATIONS UNIVERSITY MATHEMATICAL TEXTS GENERAL EDITORS ALEXANDER C. AITKEN, D.SC., F.R.S. DANIEL E. RUTHERFORD, D.SC. DR.MATH. Determinants and Matrices A. C. Aitken, D.Sc., F.R.S. Statistical Mathematics A. C. Aitken, D.Sc., F.R.S. The Theory of Ordinary Differential Equations J. C. Burkill, Sc.D., F.R.S. Waves C. A. Coulson, D.Sc., F.R.S. Electricity C. A. Coulson, D.Sc., F.R.S. Protective Geometry T. E. Faulkner, Ph.D. Integration R. P. Gillespie, Ph.D. Partial Differentiation R. P. Gillespie, Ph.D. Infinite Series J. M. Hyslop, D.Sc. Integration of Ordinary Differential Equations E. L. Ince, D.Sc. Introduction to the Theory of Finite Groups ... ... ... ... W. Ledermann, Ph.D., D.Sc. German-English S. Macintyre, Ph.D., and E. Witte, M.A. Mathematical Vocabulary Analytical Geometry of Three Dimensions W. H. McCrea, Ph.D., F.R.S. Topology E. M. Patterson, Ph.D. Functions of a Complex Variable ... E. G. Phillips, M.A., M.Sc. Special Relativity W. Rindler, Ph.D. Volume and Integral ... W. W. Rogosinski, Dr.Phil., F.R.S. Vector Methods D. E. Rutherford, D.Sc., Dr.Math. Classical Mechanics D. E. Rutherford, D.Sc., Dr.Math. Fluid Dynamics D. E. Rutherford, D.Sc., Dr.Math. Special Functions of Mathematical Physics and Chemistry I. N. Sneddon, D.Sc. Tensor Calculus* B. Spain, Ph.D. Theory of Equations H. W. Turnbull, F.R.S. FUNCTIONS OF A COMPLEX VAEIABLE WITH APPLICATIONS BY E. G. PHILLIPS, M.A., M.So. SENIOR LECTURER IN MATHEMATICS IN THE UNIVERSITY COLLEGE OF NORTH WALKS, TUNGOR: FORMERLY SENIOR SCHOLAR OF TRINITY COLLKGB, CAMBRIDGE With 17 Figure* OLIVER AND BOYD EDINBURGH AND LONDON YORK: INTERSCIENOE PUBLISHERS, INC. FIRST EDITION . . 1940 EIGHTH EDITION . . 1957 REPBINTED . . . 1961 PRINTED AND PUBLISHED IN GREAT BRITAIN BY OLIVER AND BOYD LTD., EDINBURGH PREFACE TO THE EIGHTH EDITION CHANGES that have been made in recent editions include a set of Miscellaneous Examples at the end of the book and an independent proof of Liouville's theorem has been given. In this edition, the proof of the Example on page 60 has been altered. Limitations of space made it necessary for me to confine myself to the more essential aspects of the theory and its applications, but I have aimed at including those parts of the subject which are most useful to Honours students. Many readers may desire to extend their knowledge of the subject beyond the limits of the present book. Such readers are recommended to study the standard treatises of Copson, Functions of a Complex Variable (Oxford, 1935), and Titchmarsh, Theory of Functions (Oxford, 1939). I take this opportunity of acknowledging my constant indebtedness to these works both in material and presentation. I have presupposed a knowledge of Real Variable Theory corresponding approximately to the content of my Course of Analysis (Cambridge, Second Edition, 1939). References are occasionally given to this book in footnotes as P.A. I wish to express my thanks to all those friends who have made helpful suggestions. In particular, I mention two of my colleagues, Mr A. C. Stevenson, of University College, London, who read the proofs of the first edition, and Prof. H. Davenport, F.R.S., who very kindly suggested a number of improvements for the second edition. I desire also to express my gratitude to the publishers for the careful and efficient way in which they have carried out their duties. - E Q p BANGOR, October 1956 vii CONTENTS PREFACE p. vii FUNCTIONS OF A COMPLEX VARIABLE p. 1 PAGB < Complex Numbers ....... I Sets of Points in the Argand Diagram ... 7 Functions of a Complex Variable .... 9 Regular Functions . . . . . . .11 Conjugate Functions . . . . . .14 ^JPower Series 17 The Elementary Functions . . . . .20 Many -valued Functions . . . . . .26 Examples I ........ 29 CONFORMAL REPRESENTATION p. 32 Isogonal and Conformal Transformations . . .32 Harmonic Functions . . . . . .37 The Bilinear Transformation ..... 40 Geometrical Inversion ...... 43 The Critical Points 45 Coaxal Circles 46 Invariance of the Cross-Ratio 49 Some special Mdbius' Transformations . . .51 Examples II . , . . .54 x CONTENTS SOME SPECIAL TRANSFORMATIONS p. 67 PAGH The Transformations 10 = 2". . . .68 w = z* 60 w = \/z 62 w = tan 2 (frr\/z) 64 Combinations of w == z* with Mobius' Transformations 66 Exponential and Logarithmic Transformations . . 70 Transformations involving Confocal Conies . . 71 z = c sin w ........ 74 Joukowski's Aerofoil . . . . . .76 Tables of Important Transformations . . .78 Schwarz-Christoffel Transformation .... 80 Examples III ........ 81 THE COMPLEX INTEGRAL CALCULUS p. 85 Complex Integration ...... 85 Cauchy's Theorem . . . . . . .89 The Derivatives of a Regular Function ... 93 Taylor's Theorem 95 Liouville's Theorem . . . . . .96 Laurent's Theorem . . . . . . .97 Zeros and Singularities ...... 98 Rational Functions . . . . . . .103 Analytic Continuation . . . . . .103 ' Poles and Zeros of Meromorphic Functions . .107 Rouch6's Theorem . . . . . . .108 The Maximum-Modulus Principle . . . .109 Examples IV Ill THE CALCULUS OF RESIDUES p. 114 The Residue Theorem ...... Integration round the Unit Circle 117 CONTENTS xi THE CALCULUS OF RESIDUES continued. PAGE Evaluation of Infinite Integrals . . . .119 Jordan's Lemma . . . . . . .122 Integrals involving Many -valued Functions . ,126 Integrals deduced from Known Integrals . . .128 Expansion of a Meromorphic Function . . .131 Summation of Series . . . . . .133 Examples V ........ 135 MISCELLANEOUS EXAMPLES 138 INDEX 143 OH APTBB I FUNCTIONS OF A COMPLEX VARIABLE 1. Complex Numbers This book is concerned essentially with the application of the methods of the differential and integral calculus to complex numbers. A number of the form a+ij8, where t is V(~~ 1) an( l a an d j3 are real numbers, is called a complex number ; and, although complex numbers are capable of a geometrical interpretation, it is important to give a definition of them which depends only on real numbers. Complex numbers first became necessary in the study of algebraic equations. It is desirable to be able to say that every quadratic equation has two roots, every cubic equation has three roots, and so on. If real numbers only are considered, the equation a +l = has no roots and x 3 l = has only one. Every generalisation of number first presented itself as needed for some simple problem, but extensions of number are not created by the mere need of them ; they are created by the definition, and our object is now to define complex numbers. By choosing one of several possible lines of procedure, we define a complex number as an ordered pair of real numbers. Thus (4, 3), (\/2, e), (J, TT) are complex numbers. If we write z = (x, j/), x is called the real part, and y the imaginary part, of the complex number z. (i) Two complex numbers are equal if, and only if, their real and imaginary parts are separately equal. The equation z = z' implies that both x = x' and y = y'. I 2 FUNCTIONS OF A COMPLEX VARIABLE (ii) The modulus of z, written | z |, is defined to be + \/(x*-\-y*). H> follows immediately from the definition that | z | = if, and only if , x = and y = 0. (iii) The fundamental operations. If z = (z, y), z' = (#', y') we have the following definitions : (1) z+z' is (*+*', y+y'). (2) -zis(-*, -t/). (3) z-z' = z+(-z') is (z-a;', y-y'). (4) 22' is (xx'yy' 9 xy'+x'y). If the fundamental operations are thus defined, we easily see that the fundamental laws of algebra are all satisfied. (a) The commutative and associative laws of addition hold : (b) The same laws of multiplication hold : (o) TAe distributive law holds : As an example of the method, we show that the com- mutative law of multiplication holds. The* others are proved similarly. We have thus seen that complex numbers, as defined above, obey the fundamental laws of the algebra of real numbers : hence their algebra will be identical in /orm, though not in meaning, with the algebra of real numbers. We observe that there is no order among complex numbers. As applied to complex numbers, the phrases " greater than " or " less than " have no meaning. In- equalities can only occur in relations between the moduli of complex numbers. FUNCTIONS OF A COMPLEX VARIABLE 3 (iv) The definition of division. Consider the equation z = z', where z = (x, y), = (, i)) 9 z' = (x' 9 y') t then we have so that xyr\ = z' and, on solving for and 77, _ yy'+xx' _ xy'x'y provided that | z | ? 0. Hence, if | ^ | ^ 0, there is a unique solution, and = (f , TJ) is the quotient z'/z. Division by a complex number whose modulus is zero is meaningless ; this conforms with the algebra of real numbers, in which division by zero is meaningless. The abbreviated notation. It is customary to denote a complex number whose imaginary part is zero by the real-number symbol a?. If we adopt this practice, it is essential to realise that x may have two meanings (i) the real number x, and (ii) the complex number (x, 0). Although in theory it is important to distinguish between (i) and (ii), in practice it is legitimate to confuse them ; and if we use the abbreviated notation, in which x stands for (x, 0) and y for (y, 0), then x+y = (*, 0)+(y, 0) = (x+y, 0), xy = (x, 0) . (y, 0) = (x. y-0.0, x . 0+0 . y) = (xy, 0). Hence, so far as sums and products are concerned, complex numbers whose imaginary parts are zero can be treated as though they were real numbers. It is customary to denote the complex number (0, 1) by . With this convention, t 2 = (0, 1) . (0, 1) = ( 1, 0), so that i may be regarded as the square root of the real number 1. On using the abbreviated notation, it follows that (x, y) = x+iy, FUNCTIONS OF A COMPLEX VARIABLE for, since % = (0, 1), we have = (*,0)+(0.y-1.0,0.0+l.y) = (a?, 0)+(0, y) = (a+0, 0+y) = (x, y). In virtue of this relation we see that, in any operation involving sums and products, it is allowable to treat x, y and i as though they were ordinary real numbers, with the proviso that i* must always be replaced by 1. 2. Conjugate Complex Numbers If z = x+iy, it is customary to write x = Rz, y = Iz. The number x iy is said to be conjugate to z and is usually denoted by z. It readily follows that the numbers conjugate to z l -\-z 2 and 2 t 2 2 are Zi+z^ and 2^23 respectively. Proofs of theorems on complex numbers are often considerably simplified by the use of conjugate complex numbers, in virtue of the relations, easily proved, | Z | f == 22, 2R2 = Z+Z, 2ilz = 22. To prove that the modulus of the product of two complex numbers is the product of their moduli, we proceed as follows : 2 = and so, since the modulus of a complex number is never negative, I *i*a H I *i I I * ! Theorem. The modulus of the sum of two complex numbers cannot exceed the sum of their moduli. 2 = and so FUNCTIONS OF A COMPLEX VARIABLE 5 a result which can be readily extended by induction to any finite number of complex numbers. In .a similar way we can prove another useful result, viz. K-*il>l(l*iHil)l- We have hence = (l*i H*i I) 2 ; \*i~**\ > 1(1 *i H * 1)1- 3. Geometrical Representation of Complex Numbers If we denote (# 2 +y 2 )* by r, and choose so that r cos 6 = x, r sin = y, then r and are clearly the radius FIG. 1. vector and vectorial angle of the point P, (x, y), referred to an origin and rectangular axes Ox, Oy. It is clear that any complex number can be represented geometrically by the point P, whose Cartesian coordinates are (x, y) or whose polar coordinates are (r, 0), and the representation of complex numbers thus afforded is called the Argand diagram. By the definition already given, it is evident that r is the modulus of z = (x, y) ; the angle is called the argument of z, written = arg z. The argument is not unique, for if be a value of the argument, so also is FUNCTIONS OF A COMPLEX VARIABLE 2/17T+0, (n = 0, 1 2, ...) The principal value of arg z is that which satisfies the inequalities ?r<arg Z^TT. Let P! and P 2 (in fig. 1) be the points z l and z 2 , then we can represent addition in the following way. Through P l9 draw PjP 8 equal to, and parallel to QP 2 . Then P 8 has coordinates (a^+^a* yi+^a) an d 8O -Ps represents the point z t +z 2 . In vectorial notation, QP 3 = Similarly, we have, if P 8 is the point 2 3 , It is convenient to write cis0 for cos^+fsin^. If z l = r l cis0 l9 z 2 = r 2 cis^ 2 , ..., z n = r n cis0 n , then, by de Moivre's theorem, which readily exhibits the fact that the modulus and argument of a product are equal respectively to the product of the moduli and the sum of the arguments of the factors. In particular, if n be a positive integer and z = r cis 0, z n = r n cis n$. Similarly, ^ = ^1 cis ^-fla). Z 2 r 2 If n is a positive integer, there are n distinct values of z l i*. If m is any integer, since j cis 1 = cis0, \ / it follows that r*-t n cis{(0+2w7r)/n} is an nth root of z=r cis0. If we substitute the numbers 0, 1, 2, ... n 1 in succession for m, we obtain n distinct values of z l l n ; and the sub- stitution of other integers for m merely gives rise to repetitions of these values. Also, there can be no other FUNCTIONS OF A COMPLEX VARIABLE 7 values, since z l l n is a root of the equation u n = z which cannot have more than n roots. Similarly, if p and q are integers prime to each other and q is positive, where m = 0, 1, 2, ..., ql. By considering the modulus and argument of a complex number, the operation of multiplying any complex number x+iy by i is easily seen to be equivalent to turning the line OP through a right-angle in the positive (counter-clockwise) sense. We have just seen that (- 1 ) \z % / zj) = arg Zx+arg z 2> arg - = arg z l arg z a , \z % / so that the formal process of " taking arguments " is similar to that of " taking logarithms." Hence, if arg (x+iy) = a, arg i(x+iy) = arg i+a,Tg(x+iy) = JTT+OU Since | i \ = 1, multiplying by i leaves | x+iy \ unaltered. 4. Sets of Points in the Argand Diagram We now explain some of the terminology necessary for dealing with sets of complex numbers in the Argand diagram. We shall use such terms as domain, contour, inside and outside of a closed contour, without more precise definition than geometrical intuition requires. The general study of such questions as the precise determination of the inside and outside of a closed contour is not so easy as our intuitions might lead us to expect.* For our present purpose, however, we shall find that no difficulties arise from our relying upon geometrical intuition. By a neighbourhood of a point z in the Argand diagram, we mean the set of all points z such that |z--z |<, where is a given positive number. A point z is said * For further information, see e.g. Dienee, The Taylor Seriet (Oxford, 1931), Ch. VI. 8 FUNCTIONS OF A COMPLEX VARIABLE to be a limit point of a set of points S 9 if every neighbour- hood of z contains a point of S other than z . The definition implies that every neighbourhood of a limit point z contains an infinite number of points of S. For, the neighbourhood | z z |< contains a point z 1 of S distinct from z , the neighbourhood | z z |<| z l z | contains a point z 2 of S distinct from z and so on indefinitely. The limit points of a set are not necessarily points of the set. If, however, every limit point of the set belongs to the set, we say that the set is closed. There are two types of limit points, interior points and boundary points. A limit point z of S is an interior point if there exists a neighbourhood of z which consists entirely of points of S. A limit point which is not an interior point is a boundary point. A set which consists entirely of interior points is said to be an open set. It should be observed that a set need not be eithet open or closed. An example of such a set is that consisting of the point z = 1 and all the points for which | z | <1. We now define a Jordan curve. The equation z = x(t)+iy(t), where x(t) and y(t) are real continuous functions of the real variable t, defined in the range a <J<j8, determines a set of points in the Argand diagram which is called a continuous arc. A point z x is a multiple point of the arc, if the equation z l = x(t) +iy(t) is satisfied by more than one value of t in the given range. A continuous arc without multiple points is called a Jordan arc. If the points corresponding to the values a and j3 coincide, the arc, which has only one multiple point, a double point corresponding to the terminal values a and j8 of t, is called a simple closed Jordan curve. A set of points is said to be bounded if there exists a constant K such that | z | ^.K is satisfied for all points FUNCTIONS OF A COMPLEX VARIABLE 9 z of the set. If no such number K exists the set is unbounded. A domain is defined as follows: A set of points in the Argand diagram is said to be connex if every pair of its points can be joined by a polygonal arc which consists only of points of the set. An open domain is an open connex set of points. The set, obtained by adding to an open domain its boundary points, is called a closed domain. The Jordan curve theorem states that a simple closed Jordan curve divides the plane into two open domains which have the curve as common boundary. Of these domains one is bounded and it is called the interior, the other, which is unbounded, is called the exterior. Although the result stated seems quite obvious, the proof is very complicated and difficult. When using simple closed Jordan curves consisting of a few straight lines and circular arcs, geometrical intuition makes it obvious which is the interior and which is the exterior domain. For example, the circle | z \ = R divides the Argand diagram into two separated open domains | z \<R and | z | >7?. Of these the former is a bounded domain and is the interior of the circle | z \ = R ; the latter, which is unbounded, is the exterior of the circle | z \ = R. In complex variable theory we complete the complex plane by adding a single point at infinity. This point is defined to be the point corresponding to the origin by the transformation z' = 1/2. 5. Functions of a Complex Variable. Continuity If u?(= u+iv) and z(= x-}-iy) are any two complex numbers, we might say that w is a function of z, w = /(z), if, to every value of z in a certain domain D, there correspond one or more values of w. This definition, similar to that given for real variables, is quite legitimate, but it is futile because it is too wide. On this definition, a function of 10 FUNCTIONS OF A COMPLEX VARIABLE the complex variable z is exactly the same thing as a complex function u(x, y)+iv(x,y) of two real variables x and y. For functions defined in this way, the definition of continuity is exactly the same as that for functions of a real variable. The function /(z) is continuous at the point Z Q if, given any e, >0, we can find a number 8 such that for all points z of D satisfying | z z 1 <8. The number 8 depends on e and also, in general, upon z . If it is possible to find a number h(e) independent of z , such that \f( z )f( z o) |< holds for every pair of points z, z of the domain D for which |z z |<A, then /(z) is said to be uniformly continuous in D. It can be proved that a function which is continuous in a bounded closed domain is uniformly continuous there.* It is easy to show that this definition of continuity is equivalent to the statement that a continuous function of z is merely a continuous complex function of the two variables x and y, for, if /(z) = u(x, y)+iv(x, y), when /(z) is continuous on the above definition, so are u(x, y) and v(x y y) ; and conversely, if u and v are continuous functions of x and y, /(z) is a continuous function of z. The only class of functions of z which is of any practical utility is the class of functions to which the process of differentiation can be applied. 6. Differentiability We next consider whether the definition of the derivative of a function of a single real variable is applicable * For a proof of this theorem for a closed interval, see Phillips, A Course of Analysis (Cambridge, 1939), p. 73. This will be referred to subsequently as P.A. FUNCTIONS OF A COMPLEX VARIABLE II to functions of a complex variable. The natural definition is as follows : Let f(z) be a one-valued function, defined in a domain D of the Argand diagram, then f(z) ia differentiate at a point Z Q of D if tends to a unique limit as z-^- z , provided that z is also a point of D. If the above limit exists it is called the derivative of /(z) at z = z and is denoted by /'(z ). Restating the definition in a more elementary form, it asserts that, given >0, we can find a number S such that (*)-/(*o) , Z-Z o < for all z, z in D satisfying 0< |z z |<8. That continuity does not imply differentiability is seen from the following simple example : Let/(z) = | z )'. This continuous function ia differentiable at the origin, but nowhere else. For if z ^ we havo = +z (cos 2< i sin 2<f>) where <f> = arg (z 2 ). It is clear that this expression does not tend to a unique limit as z-> z . If Zg the incrementary ratio is z t which tends to zero as z-> 0. 7. Regular Functions A function of z which is one-valued and differentiable at every point of a domain D is said to be regular * in the domain D. A function may be differentiable in a domain save possibly for a finite number of points. These points are called singularities of /(z). We next discuss the necessary and sufficient conditions for a function to be regular. * The terms analytic and holomorphic are sometimes used as ynonymoua with the term regular as defined above. 12 FUNCTIONS OF A COMPLEX VARIABLE (1) The necessary conditions for f(z) to be regular. If f(z) =s u(x, y)+iv(x, y) is differentiable at a given point 2, the ratio {/(z+ Az) f(z)}jAz must tend to a definite limit as Jz->0 in any manner. Now Az = Ax+iAy. Take A z to be wholly real, so that Ay = 0, then u(x+Ax, y)u(x y y) . v(x+Ax, y)v(x, y) Ax + * Ax must tend to a definite limit as Ax-* 0. It follows that the partial derivatives u x , v 9 must exist at the point (x 9 y) and the limit is u 9 -\-iv x . Similarly, if we take Az to be wholly imaginary, so that Ax = 0, we find that t/,, v y must exist at (x, y) and the limit in this case is v y iuy. Since the two limits obtained must be identical, on equating real and imaginary parts, we get UX = Vy , Uy = V 9 . . . (1) These two relations are called the Cauchy-Riemann differential equations. We have thus proved that for the function f(z) to be differentiable at the point z it is necessary that the four partial derivatives u x , v x , u y , v y should exist and satisfy the Cauchy- Riemann differential equations. We thus see that the results of assuming differentiability are more far-reaching than those of assuming continuity. Not only must the functions u and v possess partial derivatives of the first order, but these must be connected by the differential equations (1). That the above conditions are necessary, but not sufficient, may be seen by considering Examples 6 and 7 at the end of this chapter. (2) Sufficient conditions for f(z) to be regular. Theorem. The continuous one-valued function f(z) is regular in a domain D if the four partial derivatives u x , v xt u y , v v exist, are continuous and satisfy the Cauchy-Riemann equations at each point of D. FUNCTIONS OF A COMPLEX VARIABLE 13 Now Au = u(x+Ax y y+Ay)u(x, y), = u(x+Ax, y+Ay) u(x+Ax, y) +u(x+ Ax, y) u(x, y), = Ay . u v (x+Ax y y+OAy)+Ax . u x (x+6'Ax, y) ; where 0<0<1, 0<0'<1, by the mean- value theorem.* Since u x , u v are both continuous, we may write Au = Ax{u x (x, y)+<i}+Ay{u v (x, y)+c'} 9 where and e' both tend to zero as | Az |-> 0. Similarly, Av = Ax{v x (x, y)-i- r rj}+Ay{v 1/ (x, y)+^'}, where rj and rf both tend to zero as | Az |-> 0. Hence Aw = Au+iAv = AX(UB +iv x ) +Ay(u v +iv v ) +a)Ax+a)'Ay 9 where a* and aj f tend to zero as | Az \-> 0. On using the Cauchy-Riemann equations we get Aw = (Ax+iAy)(u x +iv x )-\-a>Ax+a>'Ay and, on dividing by Az and taking the limit as \Az |-> 0, dw dz atAx+oj'Ay . . . since ^ I co I ~p I co I We notice that the above sufficient conditions for the regularity of f(z) require the continuity of the four first partial derivatives of u and v. If w = tt-f-iv, where u and v are functions of x and y, since ..!,., -!,-* * See P.A., p. 101. 14 FUNCTIONS OF A COMPLEX VARIABLE u and t; may be regarded formally as functions of two independent variables z and z. If u and v have continuous first-order partial derivatives with respect to x and y, the condition that w shall be independent of z is that dw/8S = 0. This leads to the result du dx that is du dy /<9v dx dv dy\ __ 8y'8i\8x'di~dy8s)~~ ' I du 1 du t dv 1 dv __ ^ and, on writing f for 1/f and equating real and imaginary parts, we get du dv dv du which are the Cauchy-Riemann equations. Hence, in any analytical formula which represents a regular function of z, x and y can occur only in the combina- tion x+iy. For example, it is clear at a glance that sin (x+3iy) = sin (2zz) cannot be a regular function. If u+iv =f(x-\-iy) where f(z) is a regular function, then the real functions u and v of the two real variables x and y are called conjugate functions. Since the partial derivatives of u and v are connected by the relations du _ 8v 8v _ _te a5~V8i~~V () if the derivatives concerned are assumed to exist and satisfy the relation (f> Xv = </> vX , it follows by partial differentiation that _ _ "" "" an FUNCTIONS OF A COMPLEX VARIABLE 15 Hence both u and v satisfy Laplace's equation in two dimensions This equation occurs frequently in mathematical physics. It is satisfied by the potential at a point not occupied by matter in a two-dimensional gravitational field. It is also satisfied by the velocity potential and stream function of two-dimensional irrotational flow of an incompressible non-viscous fluid. By separating any regular function of z into its real and imaginary parts, we obtain immediately two solutions of Laplace's equation. It follows that the theory of functions of a complex variable has important applications to the solution of two-dimensional problems in mathe- matical physics. It also follows from equations (1) that The geometrical interpretation of (2) is that the families of curves in the (x, y)-plane, corresponding to constant values of u and v, intersect at right angles at all their points of intersection. For if u(x, y) = c l9 then du = 0, and so 8 dx + ^dy = 0. . . . (3) ox oy Similarly, if v(x, y) = c 2 , we have The condition that these families of curves intersect at right angles is * 16 FUNCTIONS OF A COMPLEX VARIABLE where the suffixes 1 and 2 refer to the u and v families respectively. On using (3) and (4), it is easy to see that (5) reduces to (2). It is possible to construct a function /(z) which has a given real function of x and y for its real or imaginary part, if either of the given functions u(x, y) or v(x, y) is a simple combination of elementary functions satisfying Laplace's equation. A very elegant method of doing this is due to Milne-Thomson.* Since x = g (2+2), y = - (z-z), zz . (z+z zz We can look upon this as a formal identity in two inde- pendent variables z, z. On putting z = z we get Now /'(z) = UX+WB = UxiUy by the Cauchy-Riemann equations. Hence, if we write <f>i(x t y) and <f>%(x 9 y) for u 9 and u y respectively, we have /'(z) ^^(z, y)-^ 2 (* y)=<f>i(*> 0)-i^ 2 (z, 0). On integrating, we have i(*> 0)-^ 2 (z, 0)}dz+C, where C is an arbitrary constant. Similarly, if v(x, y) is given, we can prove that I 2 (z, 0)}dz+C, where ift^x, y) = v y and 2 (#, y) = v 9 . Math. Gazette, xxi. (1937), p. 228. See also Misc. Ex. !, p. 138. FUNCTIONS OF A COMPLEX VARIABLE 17 As an example, suppose that u(x,y) = e*(x ooa y y sin y), du Here fa = = e*(x cos y y sin y-fcos y), c/a? ti ^ t = ~- = e*(x sin t/ sin yy cos y). <?2/ Hence f'(z) = ^(z, 0)-t> 2 (z, 0) = e (2+!), and so f(z) = J e (s+l)efa+(7 = ze +O. 8. Power Series. The Elementary Functions 00 00 Consider the series S a n z n or 2 a n (z z ) w , where the n=0 n coefficients a n and z, z may be complex. Since the latter series may be obtained from the former by a simple change of origin, the former may be regarded as a typical power series. It is assumed that the reader is already familiar with the theory of real power series.* So far as absolute convergence is concerned, everything that has been proved for absolutely convergent series of real terms extends at once to complex series, for the series of moduli Kl +lilN + KIN 2 +- is a series of positive terms. The most useful convergence test for power series is Cauchy's root test, which states that a series of positive terms Zu n is convergent or divergent according as liin (u n ) l l n is less than or greater than uriity.f If we write lim | a n \ l l n = 1/jR, then we easily see that the power series Za n z n is absolutely con- vergent if | z \<R, divergent if | z |>J2, and if | z | = R we can give no general verdict and the behaviour of the series may be of the most diverse nature. The number B is called the radius of convergence, and the circle, centre the origin, and radius R, is called the circle ol * See P.A., Ch. XIII. t See P.A., p. 124. 18 FUNCTIONS OF A COMPLEX VARIABLE convergence of the power series. Clearly there are three cases to consider (i) R = 0, (ii) R finite, (iii) R infinite. The first case is trivial, since the series is then convergent only when 2 = 0. In the third case the series converges for all values of z. In the second case the radius of the circle of convergence is finite and the power series is absolutely convergent at all points within this circle, and divergent at all points outside it. We now prove an important theorem. 00 If f(z) = S a n z n , then the sum-function f(z) is a regular o function at every point within the circle of convergence of the power series. Suppose that Sa n z" is convergent for | z \<R. Then, if 0</><J?, a^ n is bounded, say |anp n |</f. Let $(z) = 2 na n z*~ l . n-l We write, for convenience, | z \ = r, | h \ = 77 : then, if r<p and n-o ^-| -nan-* . JJ Now -. = (r+^-r* _ ,-! Hence /(+*)-/(*) A #(*) < KZ 1 -( (r ^ )n - rn -nr->} n-op n ( -n } l -(-r P-} /L_l r) \p-r-r) p-rj (p-r) 2 J FUNCTIONS OF A COMPLEX VARIABLE 19 which tends to zero as TJ-> 0. Hence f(z) has the derivative (f)(z). This proves that /(z), which is plainly one-valued, is also differentiable : hence f(z) is regular within | z \ = B. Since n l l n -*\ as n->oo, lim|n^ tt | 1 / n = limlaj 1 /" = 1/J2, and so the series <f>(z) = Z na n z n ~ l has the same radius of n-l convergence as the original series. Thus, if <f>(z) = /'(z) is regular in | z \<R, we can show similarly that its derivative is Sn(n l)a n z n ~ 2 , and so on. In other words, we thus prove that a power series can be differentiated term by term as often as we please at any point within its circle of convergence. The above theorem, which is the analogue of a well-known theorem in real variable theory, can be superseded by a more general theorem which is one of the characteristic achievements of complex variable theory. This theorem is as follows : Let f(z) _ u l (z)+ui(z) + . . . +u n (z)+ . . . ; if each term u n (z) is regular within a region D and if the series is uniformly convergent throughout every region D' interior to D, then f(z) is regular within D and all its derivatives may be calculated by term-by -term differentiation. For a proof of this theorem, the reader is referred to larger treatises on complex variable theory. The simpler theorem proved above will suffice for the purposes of this book. In a later chapter ( 34) we prove Taylor's theorem that a function /(z) can be expanded in a power series 00 2a n (za) n about any point a, provided that/(z) is regular n-O in | za |<p. By combining Taylor's theorem with the theorem proved above, we see that the necessary and sufficient condition that a function /(z) may be expanded in a power series is that it should be regular in a region. The Weierstrassian development of complex variable 20 FUNCTIONS OF A COMPLEX VARIABLE theory begins by defining an " analytic function " of z aa a function expansible in a power series. (See 39.) We now consider briefly the definitions of the so-called elementary functions of a complex variable. I. Rational functions. A polynomial in z, a Q -\-a l z+...+a m z m 9 may be regarded as a power series which converges for all values of z. Since such functions are regular in the whole plane, rational functions of the type are regular at all points of the plane at which the denominator does not vanish. If we choose a point z , at which the denominator does not vanish, and replace * by Z +(z--z ), the function /(z) becomes A +A l (z-z )+...+A rn (z-z )< in which B Q ^ 0. It readily follows that /(z) may be 00 expanded in a power series of the form Hc n (zz ) n . o II. The exponential function. For the exponential function of a real variable, one method of development is to define exp a: as the sum- function of the power series and, on using the multiplication theorem for absolutely convergent series, we prove that * exp x . exp x 9 = exp (x+x'). * See P.A., p. 246. FUNCTIONS OF A COMPLEX VARIABLE 21 In the same way we can define eipz as the sum- function of the series of complex terms Since the series converges for all values of z, it defines a function regular in the whole z-plane. Such functions are called integral functions. When x is rational, exp x is identical with the function e* of elementary algebra, and when x is irrational we define e 9 to be identical with the function exp x, the sum-function of the power series (1) above. In the same way, when z is complex, we find it convenient conventionally to write e* for exp z. Since the formula exp z . exp = exp(z + f) can be proved by multiplication of series, whether z be real or complex, the real number e with a complex exponent obeys the formal law of indices of elementary algebra e'. ef = e*+f . Thus we may define the power e*, without ambiguity, by the equation z 2 z 8 e = l + z +2\ + 3! + * and, if a is any positive number, a* denotes the value unambiguously determined by the formula a * ^ gi log*, where log a is the real natural logarithm of a. The reader should notice how far this definition is removed from the elementary definition " x k is the product of k factors equal to a?." At first sight there is no knowing what value belongs to a number of the form 2', but its value is uniquely determined by our definition. 22 FUNCTIONS OF A COMPLEX VARIABLE For real values of y we have 00 (iij} n & f/2* oo e" - 27 ( -ff - 2; (-1)* ITT-, + t(-l) --y- (2) n-o * ! t o ( 2k ) ! *~o (2&+1) ! = cos y+ i sin y ; since the cosine and sine of the real variable y are defined by the two power series on the right of (2). Hence e* = e** iv = eV* = e* cis y. We also see that, since | & | = | cisy | = 1 , | e | = | e|| e<* \ = e, since e*>0. Similarly, arg e f = Iz = y. TAe function e 9 has the period 2ni ; in other words, if k is any positive or negative integer, or zero, for, when we increase z by 27rt, y increases by 2?r and this leaves the values of sin y and cos y unchanged. Every value which e* is able to assume is therefore taken in the infinite strip 7r<j/<7r, or in any strip obtainable from this by a parallel translation. It is easy to show that e 9 has no other period. If e f = ef, this necessarily implies that z = f +2kiri. This follows at once, because e f ~f = 1 and so e*~f cis(y 77) = 1. Hence x % = 0, cos (yrj) = 1, sin (yrj) = ; and this leads to y7) = Zkn, so that z = 2&?rt. Finally, e* never vanishes, for e**e-*i = 1, and, if e* =s 0, this equation would give an infinite value for e~*>, which is impossible. Since z = z+iy = r cos 0+ir sin 0, any complex number may be written in the form z = re^, where | z \ = r, arg z = #, since we have now assigned a meaning to e'0. FUNCTIONS OF A COMPLEX VARIABLE 23 By term-by-term differentiation of the power series defining e* t we readily see that T *>* = * dz III. The trigonometrical and hyperbolic functions. We define sin z and cos z, when z is complex, as the sum-functions of power series, just as we do for sin x and cos x when x is real. Thus sin Z = - (2n+l)I* and, since each of these power series has an infinite radius of convergence, sin z and cos z are integral functions. By term-by-term differentiation of these power series, we deduce at once that the derivatives of sin z and cos z are cos z and sin z respectively. The other trigonometrical functions are then defined by sin z A 1 1 1 tan z = , cot z = , sec z = , cosec z = - . cosz tanz cosz sinz If we denote exp iz by e**, according to our agreed con- vention, we readily obtain the results cos z+i sin z = e**> cos z i sin z == er** ; leading to Euler's formulae cos z = - (e' +*-*) , sin z = - (et'-e-**). & +->i From these formulae, and the addition formula for e*, we find that . , , sm 2 z+cos 2 z =5 1 ; and the addition theorems sin(z ) = sin z cos cos z sin , cos(z J) s=s cos z cos S^sin z sin , also hold for complex variables. As all the elementary identities of trigonometry are algebraic deductions from 24 FUNCTIONS OF A COMPLEX VARIABLE these fundamental equations, all such identities also hold for the trigonometrical functions of a complex variable. The hyperbolic functions of a complex variable are also defined in the same way as for real variables. The two fundamental ones, from which the others may be derived, are sinh z = |(e f e~*), cosh z = \(e* +e~ f ). These two functions are clearly regular in any bounded domain. The important relations sin iz = i sinh z, cos iz = cosh , sinh iz = i sin z, cosh iz = cos z, are easily proved and are of great usefulness for deducing properties of the hyperbolic functions from the correspond- ing properties of the trigonometrical functions. If we write z = x+iy, sin z = sin x cosh y +i cos x sinh y, and we see that sin z can only vanish if sin x cosh y = 0, cos x sinh y = 0. Now coshy>l, and so the first equation implies that sin x is zero. Hence x = mr, (n = 0, 1, ^2, ...). The second then becomes sinh y = 0, and this has only one root y = 0. Hence sin z vanishes if, and only if, z = ttTr, (n = 0, 1, 2, ...). Similarly, we can show that cos z vanishes if, and only if, z = (n-\-%)ir. IV. The logarithmic function. When x is real and positive, the equation e" = x has one real solution u = log x. If z is complex, however, but not zero, the corresponding equation exp w = z has an infinite number of solutions, each of which is called a logarithm of z. If w = u+iv we have e"(cos t>+*' sin t;) = z. FUNCTIONS OF A COMPLEX VARIABLE 25 Hence we see that v is one of the values of arg z and e u = \z\. Hence u = log | z |. Every solution of exp w = z is thus of the form w = log |z|+iargz. Since arg z has an infinite number of values, there is an infinite number of logarithms of the complex number z, each pair differing by 2iri. We write Log z = log \z | + i arg z, so that Log z is an infinitely many- valued function of z. The principal value of Log z, which is obtained by giving argz its principal value, will be denoted by logz, since it is identical with the ordinary logarithm when z is real and positive. We refer again to the logarithmic function in the next section, where many-valued functions are discussed in more detail. V. The general power f . So far we have only defined a 1 when a>0. If z and denote any complex numbers we define the principal value of the power *, with ^ as the only condition, to be the number uniquely determined by the equation where log is the principal value of Log . By choosing other values of Log we obtain other values of the power which may be called its subsidiary values. All these are contained in the formula Hence f has an infinite number of values, in general, but one, and only one, principal value. Example, t* denotes the infinity of real numbers exp{i(logi+2kiri)} = exp {i(%7Ti+2kiri)} = exp ( ^TrZk-rr). exp (Jw) is the principal value of the power i 4 . If 0, Rz>0, we define f to be zero. 26 FUNCTIONS OF A COMPLEX VARIABLE 9. Many-valued Functions In the definition of a regular function given in 7, we note that a regular function must be one-valued (or uniform). Quite a number of elementary functions, such as 2 (a not an integer) log z, arc sin z are many- valued. To illustrate the idea of many-valuedness, let us consider the simple case of the relation w 2 = z. On putting z = re*Q, w = Re i( t> 9 we get For given r and 9(<2rr), two obvious solutions are Wl = | \/r | e** 8 and w 2 = | y> I e^ e +^ = -| y> I* 4 **, and these are the only continuous solutions for fixed 0, since |yV | and | \/r \ are the only continuous solutions of the real equation x 2 = r, r>0. In particular, for a positive real z, that is when = 0, w = | \/r | and u> 2 = | \/ r l> an d both w l and tu 2 are one-valued functions of z defined for all values of z. If we follow the change in w l as varies from to 27r, in other words, as the variable z describes a circle of radius r about the origin, w l varies continuously and we see that the final value of w l is | \Jr \ e^ ' 27rt = | ^/r \ = w 2 . Hence the function w l is apparently discontinuous along the positive real axis, since the values just above and just below the real axis differ in sign and are not zero (except at the origin itself). If, however, z describes the circle round the origin a second time, the values of w l continue those of w 2 and at the end of the second circuit we have w 2 = w l along the positive real axis. We thus see that the equation w 2 = z has no continuous one-valued solution defined for the whole complex plane, but w 2 = z defines a two-valued function of z. The two functions w l = | \/r \ e W and w? 2 = | \/r \ e^ are called the two branches of the two-valued function w 2 = z. Each of these branches is a one- valued function FUNCTIONS OF A COMPLEX VARIABLE 27 in the z-plane if we make a narrow slit, extending from the origin to infinity along the positive real axis, and distinguish between the values of the function at points on the upper and lower edges of the cut. If OA = x , in fig. 2, the value of w l at A(6 = 0) is | \/# | and the value B * FIG. 2. of w l at B(6 = 2n) is | \/a? I- Since the cut effectively prevents the making of a complete circuit about the origin, if we start with a value of z belonging to the branch w v we can never change over to the branch w%. Thus w l (and similarly u> 2 ) is one-valued on the cut-plane. There is an ingenious method of representing the two- valued function w 2 = z as a one-valued function, by constructing what is known as a Riemann surface. This is equivalent to replacing the ordinary z-plane by two planes P l and P 2 : we may think of P l as superposed on P 2 . If we make a cut, as described above, in the two planes, we make the convention that the lower edge of the cut in P l shall be connected to the upper edge of the cut in P 2 and the lower edge of the cut in P 2 to the upper edge of the cut in P v Suppose that we start with a value z of z, t#J being the corresponding value of w l on the plane P 1 , and let the point z describe a path, starting from z , in the counter-clockwise sense. When the moving point reaches the lower edge of the cut in Pj it crosses to the upper edge of the cut in P 2 , then describes another counter-clockwise circuit in P 2 until it reaches the lower edge of the cut in this plane. It then crosses again to the upper edge of the cut in P 1 and returns to its starting point with the same value W with which it started. This corresponds precisely to the way in which we obtain the two different values of \/z, and so \/z is a one-valued function of position on the Riemann surface. 28 FUNCTIONS OF A COMPLEX VARIABLE There is no unique way of dividing up the function into branches, and we might have cut the plane along any line extending from the origin to infinity, but the point z = is distinguished, for the function w = \/z 9 from all other points, as we shall now see. We observe that, if z describes a circle about any point a and the origin lies outside this circle, then arg z is not increased by 2n but returns to its initial value. Hence the values of w l and u> 2 are exchanged only when z turns about the origin. For this reason the point z = is called a branch-point of the function w = \/z and, as we have already stated, w^z) and w 2 (z) are called its two branches. Since turning about z = oo means, by definition, describing a large circle about the origin, the point z = oo is also a (conventional) branch-point for w = <\/z. The relation w n = z defines an n- valued function of z, i since z* has n, and only n, different values 87T n , (s = 0, 1, 2, ..., ft 1). The point z = is a branch-point, and the Riemann surface appropriate to this function consists of n sheets P 19 P 2 , ..., P n - Plainly z = 1 is a branch-point for w = \/(z 1) and the cut is made from z = 1 to z = oo. For w = Log z, since w = log r+i(0-f 2for), every positive and negative integer k gives a branch, so Log z is an infinitely many- valued function of z. The Riemann surface consists of an infinity of superposed planes, each cut along the positive real axis, and each edge of each cut is joined to the opposite edge of the one below. The points z = and z = oo are branch points. For w = <\/{(za)(zb)} we make a cut on each plane along the straight line joining the points z = a and z = 6, and join the planes P v and P f cross- wise along the cut. In this case infinity is not a branch-point. If u>= \/{(*-~ a i)(*~ a i)---( z "~ a *)}; when k is even, FUNCTIONS OF A COMPLEX VARIABLE 29 we make cuts joining pairs of points a r , a f ; and when k is odd, one of these points must be joined to oo, as in the case k = 1. The edges of the various cuts on the two planes are joined cross- wise as in the case w = ^/z. Considerable ingenuity is required in constructing Riemann surfaces for more complicated functions, but it is beyond our scope to pursue this question further. Note on notation. In what follows we shall frequently use w, z and to denote complex numbers, and, whenever they are used, it will be understood, without further explanation, that w = u+iv, z SB x+iy, ss +1*77. Other symbols, such as t, r, s, may occasionally be used to denote complex numbers, but we do not specify any special symbols to denote their real and imaginary parts. EXAMPLES I 1. Prove that | z l -z 9 1 2 + | z^z 9 | = 2\ z l \*+2 \ z 2 | ; and deduce that all the numbers concerned being complex. 2. Prove that the area of the triangle whose vertices are the points z l9 z s , z 3 on the Argand diagram is {(* 2 -* 8 )|Zl |*/*l}. Show also that the triangle is equilateral if 3. Determine the regions of the Argand diagram defined by |s-3|<l; \z-a\ + \z-b\^k(k>0); \ z*+az+b \<r*. In the last case, show that, if z l9 z a are the roots of z*+az+b = 0, we obtain two regions if r< J | z l z t |. 4. A point P(a+ib) lies on the line AB, where A is z = p and B is z = 2ip. If Q is p a /(a +ib), find the polar coordinates 30 FUNCTIONS OF A COMPLEX VARIABLE of P and Q referred to the origin as pole and the real axis as initial line. Indicate the positions of P and Q in an Argand diagram. If P move along the line AB and G is the point z = p, prove that the triangles OAP, OQC are similar, and that the locus of Q is a circle. 5. If /<) - ' <**0),/(0) = 0, prove that {/(z) /(0)}/z-> as z-$~ along any radius vector, but not as z-> in any manner. 6. Prove that the function u+iv = /(z), where is continuous and that the Cauchy-Riemann equations are satisfied at the origin, yet/'(0) does not exist. 7. Show that the function /(z) = \f\ xy \ is not regular at the origin, although the Cauchy-Riemann equations are satisfied at that point. 8. If /(z) is a regular function of z, prove that 9. If w ^/(z) is a regular function of z such that/'(z) ^ 0, prove that If \f'(z) \ is the product of a function of x and a function oft/, show that f(z) where a is a real and )3 and y are complex constants. 10. Prove that the functions (i) u t (ii) u = sin a; cosh y +2 cos a; sinh y -\-x* t/ f both satisfy Laplace's equation, and determine the corre* spending regular function u+iv in each caae, FUNCTIONS OF A COMPLEX VARIABLE 31 11. If w = arc sin z, show that w = nrr ii according as the integer n is even or odd, a cross-cut being made along the real axis from 1 to oo, and from oo to 1 to ensure the one-valuedness of the logarithm. 12. If w =* V{(1-*)(1 +**)>, ^ the point (2, 0) and P a point in the first quadrant, prove that, if the value of w when z = is 1, and z describes the curve OP A, the value of w at A is i\/5. 13. If w = V(2 2z+z f )> and z describes a circle of centre z = 1 +i and radius V2 in the positive sense, determine the value of w (i) when z returns to O, (ii) when z crosses the axis of y, given that z starts from O with the value + \/2 of w. 14. Prove that log(l +z) is regular in the z-plane, cut along the real axis from oo to 1, and that this function can be expanded in a power series z* z* z* '-2 + 3-4 + convergent when | z \ < 1. 15. Prove that the function -"< n+1) , n~l "I is regular when | z \ < I and that its derivative is o/(z)/(l -fa). Hence deduce that/(z) = (l+z) a . 16. (i) Prove that the exponential function e 9 is a one- valued function of z. (ii) Show that the values of z = a*", when plotted on the Argand diagram for z, are the vertices of an equiangular polygon inscribed in an equiangular spiral whose angle is independent of a. 17. If 0<a <a 1 <...<a ll , prove that all the roots of the equation lie outside the circle | z | = 1. 18. Show that, if 6 is real and sin sin <f> = 1, then where n is an integer, even or odd, according as sin 0>0 or sin 0<0. [If ^ = o+t'0, we have sin a cosh ft = cosec 0, cos a sinh = 0. Solve for a and ft.] CHAPTER II CONFORMAL REPRESENTATION 10. Isogonal and Gonformal Transformations The equations u = u(x, y), v = v(x, y) may be regarded as setting up a correspondence between a domain D of the (x, t/)-plane and a domain D f of the (u, v)-plane. If the functions u and v are continuous, and possess continuous partial derivatives of the first order at each point of D t then any curve in D, which has a continuously turning tangent, corresponds to a curve in D' possessing the same property, but the correspondence between the two domains is not necessarily one-one. For example, if u = x* 9 v = t/*, the circular domain f +2/ f <l corresponds to the triangle formed by the lines u = 0, v = 0, w+v = 1, but there are four points of the circle corresponding to each point of the triangle. If two curves in the domain D intersect at the point P, (X Q , y ) at an angle 0, then, if the two corresponding curves in D' intersect at the point (U Q , v ) corresponding to P at the same angle 0, the transformation is said to be isogonal. // the sense of the rotation as well as the magnitude of the angle is preserved, the transformation is said to be conf ormal. Some writers do not distinguish between isogonal and conf ormal, but define conformality as the preservation of the magnitude of the angles without considering the sense. If two domains correspond to each other by a given transformation u = u(x, y), v = v(x 9 y), then any figure in D may be said to be mapped on the corresponding figure n CONFORMAL REPRESENTATION 33 in D' by means of the given transformation. We have already defined isogonal and conformal mapping, but it should be observed that, if one domain is mapped isogonally or conformally upon another, the correspondence between the domains is not necessarily one-one. If to each point of D there corresponds one, and only one, point of D', and conversely, the mapping of D on Z)', or of D 9 on Z), is said to be one-one or biuniform. Suppose that w = f(z) is regular in a domain D of the z-plane, z is an interior point of Z), and C l and C 2 are two continuous curves passing through the point z , and let the tangents at this point make angles a l9 a a with the real axis. Our object is to discover what is the map of this figure on the uj-plane. For a reason which will appear in a moment, we suppose thatf(z Q ) ^ 0. Let Zj and z a be points on the curves C l and (7 a near to z and at the same distance r from z q , so that z = re'i , z a z c = then, as r-> 0, 1 -> a l and 2 -> a a . The point z corresponds to a point t0 in the u?-plane and z l and z 2 correspond to points w l and u? a which describe curves /\ and F 2 . Let then, by the definition of a regular function, and, since the right-hand side is not zero, we may write it JRe<A. We have and BO lim (^j QI) == A or lim <f> l = a x -f A. 34 FUNCTIONS OF A COMPLEX VARIABLE Thus we see that the curve F l has a definite tangent at WQ making an angle 04 +A with the real axis. Similarly, F% has a definite tangent at u> making an angle a 2 +A with the real axis. It follows that the curves F l and F 2 cut at the same angle as the curves C l and C 2 . Further, the angle between the curves F l9 F^ has the same sense as the angle between O f 1 , (7 2 . Thus the regular function w=f(z), for which /'(z ) ^ 0> determines a conformal transformation. Any small figure in one plane corresponds to an approximately similar figure in the other plane. To obtain one figure from the other we have to rotate it through the angle A = arg (f'(z Q )} and subject it to the magnification It is clear that the magnification is the same in all directions through the same point, but it varies from one point to another. If is a regular function of w and w is a regular function of z, then is a regular function of z, and so, if a region of the z-plane is represented conformally on a region of the w-plane and this in its turn on a region of the -plane, the transformation from the z-plane to the {-plane will be conformal. There exist transformations in which the magnitude of the angles is conserved but their sign is changed. For example, consider the transformation w = x iy ; this replaces every point by its reflection in the real axis and so, while angles are conserved, their signs are changed. This is true generally for every transformation of the form . (I) CONFORMAL REPRESENTATION 35 where f(z) is regular ; for it is a combination of two transformations (i){~, (ii) * In (i) angles are conserved but their signs are changed, and in (ii) angles and signs are conserved. Hence in the given transformation, angles are conserved and their signs changed. Thus (I) gives a transformation which is isogonal but not conformal. We have seen that every regular function t0=/(z), defined in a domain in which /'(z) is not zero, maps the domain in the z-plane conformally on the corresponding domain in the u;-plane. Let us now consider the problem from the converse point of view. Given a pair of differenti- able relations of the type u = u(x 9 y), v = v(x, y) . . (2) defining a transformation from (x y y) -space to (u, v)-space does there correspond a regular function w = /(z) ? Let da and ds be elements of length in the (u, v) -plane and (x, y) -plane respectively. Then da 1 = du z -{-dv 2 9 ds 2 = dx^+dy 2 and so, since du du 8v dv du = dx + ~ dy , dv = dx + dy ; 8x dy 8x 8y da 2 = Edx*+2Fdxdy+Gdy\ where l-Y F=- 8 - + - \dx) ' 8x By "*" 8x 8y ' E = 4- dx Then the ratio do : ds is independent of direction if E F O 36 FUNCTIONS OF A COMPLEX VARIABLE On writing A* for E (or Q), where h depends only on x and y and is not zero, the conditions for an isogonal transformation are '- 1 * (*)'-* du du dv 8v _ 'dx dy ~8x 8y ~~ The first two equations are satisfied by writing u 9 = h cos a, v m = h sin a, u y = A cos /?, v, = A sin , and the third is plainly satisfied if Hence the correspondence is isogonal if either (a) u 9 = v v , v x = u v or (b) u m = v v , v x = u v . Equations (a) are the Cauchy-Riemann equations and express that u+iv = f(x-{-iy) where f(z) is a regular function of z. Equations (b) reduce to (a) by writing v for v, that is, by taking the image figure found by reflection in the real axis of the u;-plane. Hence (b) corresponds to an isogonal, but not conformal trans- formation, and so it follows that the only conformal transformations of a domain in the z-plane into a domain of the u?-plane are of the form w = /(z) where f(z) is a regular function of z. The casef'(z) = 0. We laid down above the condition that /'(2 ) = 0. Suppose now that/'(z) has a zero * of order n at the point z 0f then, in the neighbourhood of this point, where a = 0. Hence or Pl f "* = | a | r n + l See 36, CONFORMAL REPRESENTATION 37 where A = arg a. Hence Similarly, lim< 2 = A+(n+l)a 2 . Thus the curves F l9 F 2 still have definite tangents at w , but the angle between the tangents is Also the linear magnification, lim (pjr) 9 is zero. Hence the conformal property does not hold at such a point. For example, consider w = z*. In this arg w = 2 arg z and the angle between the line joining the origin to the point t# and the positive real axis is double the angle between the line joining the origin to the corresponding point z and the positive real axis in the z-plane. Corresponding angles at the origins are not equal because, at z = 0, dwjdz = 0. Points at which dw/dz = or oo will be called critical points of the transformation defined by w =/(z)- These points play an important part in the transformations. 11. Harmonic Functions Solutions of Laplace's equation, V 2 F = 0, are called harmonic functions ; and, in applications to mathematical physics, an important problem to be solved is that of finding a function which is harmonic in a given domain and takes given values on the boundary. This is known as Dirichlet's problem. In the three-dimensional case, if we make a transformation from (#, y, z)-space to (, T?, )-space, it will, in general, alter Dirichlet's problem. If F(x, y, z) is harmonic, and we make the transformation * = <i( >?> y = < a ( *n> 0. ^ = <&,(, 7), )> the function F 1 (^ : , 77, f ), into which F(z, y, z) is transformed is not, in general, harmonic in (, TJ, )- 8 P ace - In two- dimensional problems, however, if the transformation is 38 FUNCTIONS OF A COMPLEX VARIABLE conformal, Laplace's equation in (x, y)-space corresponds to Laplace's equation in (u, v)-space, and the problem to be solved in (u, v)-space is still Dirichlet's problem. To prove this, consider a transformation u = u(x, y), v = v(x, y) . . . (1) where w = u+iv is a regular function of z = x+iy, say w = /(z). Let Z) be a domain of the (x, y) -plane throughout which /'(z) T 0, and let J be the domain of the (u, v)-plane which corresponds to D by means of the given trans- formation. If x and y are the independent variables, and V any twice-differentiable function of x and y, we have * 8 2 V 8 2 V 8 2 V 8V 8V 8u . 8u _ . 8v . , 8v . and du = dx + dy, dv = dx + dy, . (3) ox cy ox oy . . (4) with a similar expression for d 2 v. On substituting for du, dv, d z u, d 2 v the expressions (3) and (4), the expression (2) for d 2 V becomes a quadratic expression in the differentials of the independent variables dx and dy and, on selecting the coefficients of dx 2 and dy 2 9 we get 4. 2 i a/ "*" ^\ f a. 2 + .. dy dy av 2 \8y du dy* dv dy*' But we have already seen that u and v satisfy the Cauchy- Riemann equations du __ dv dv _ 8u lfa~ 8y' fa~~ ~"8y 9 * See P.A., p. 233. CONFORMAL REPRESENTATION 39 and also Laplace's equation. On addition we therefore get 8 2 F 8 2 F\ M 2 l J ' ;rr + ^~r = 3 2 F A ^r + -T-T = 0. Now f'(z) = tt+u;, and so the last bracket is equal to |/'(z)| 2 , the square of the linear magnification of the transformation. Since this is not zero, it follows that if then 12. Superficial Magnification We have already seen that the linear magnification at any point in a conformal transformation w = f(z) is |/'(z)|, it being supposed that f'(z) ^0. We now prove that the superficial magnification is |/'(2)| 2 . If A be the closed domain of the uj-plane which corresponds to a closed domain D of the z-plane, the area A of A is given by '-///"*-//. dxdy, by the well-known theorem for change of variables in a double integral.* Now d(u, v) _ dtt dv dv 8u _ d(x, y) ~~ 8x By 8x dy ~~ by the Cauchy-Riemann equations : but, as we have just seen, du . dv 2 dx 8x and so m C C , , X|9J , \f'(z)\*dxdy. dx) \8y) I/'WI 1 This proves the theorem. * P.A., p. 302; or Gillespie, Integration, p. cited as Q.I.) 40. (Hereafter 40 FUNCTIONS OF A COMPLEX VARIABLE 13. The Bilinear Transformation We have seen that a regular function wf(z), for which /'(z ) 7^ 0, gives a continuous one-one representation of a certain neighbourhood of the point 2 of the z-plane on a neighbourhood of a point W Q of the u?-plane, and that this representation is conformal. It may be expressed in another way by saying that by such a transformation infinitely small circles of the z-plane correspond to infinitely small circles of the t0-plane. There are, however, non- trivial conformal transformations for which this is true of finite circles : these transformations will now be investigated. Let A, J3, G denote three complex constants, A, 5, their conjugates and 2, a complex variable and its conjugate ; then the equation (A+I)zz+Bz+Bz+C+D = Q . . (1) represents a real circle or a straight line, provided that BB>(A+A)(C+C). ... (2) For, if we write A = a+ia' 9 B = b+ib', C~c+ic', z = x+iy, (1) becomes a(&+y*)+bxb'y+e = which is the equation of a circle. It reduces to a straight line if a = \(A +A) = 0. If r be the radius of this circle, _ 4a a 4a 2 a' and the circle is real provided that which is the same as condition (2). Conversely, every real circle or straight line can, by suitable choice of the constants, be represented by an equation of the form (1) satisfying the condition (2). CONFORMAL REPRESENTATION 41 The transformation "-jSl < 3 > where a, j8, y, 8 are complex constants is called a bilinear transformation. It is the most general type of trans- formation for which one and only one value of z corresponds to each value of w, and conversely. Since the bilinear transformation (3) was first studied by Mobius (1790-1868) we shall, following Carath6odory,* call it also a M6bius' transformation. The expression a8 j8y, called the determinant of the transformation, must not vanish. If aS /?y = 0, the right-hand side of (3) is either a constant or meaning- less. For convenience it is sometimes agreed to arrange that aS /?y = 1. The determinant in the general case can always be made to have the value unity if the numerator and denominator of the fraction on the right- hand side of (3) be divided by \/( a <$~~/ty)- If we write ^ = (a/y) +(j8y-a8)(a>/y), a> = l/, { = yz+8 . (4) it is easily seen that (3) is equivalent to the succession of transformations (4). Now w = z+a corresponds to a translation, since the figure in the w?-plane is merely the same figure as in the z-plane with a different origin. Consider next w = pz, where p is real. The two figures in the z-plane and the w-plane are similar and similarly situated about their respective origins, but the scale of the w-figure is p times that of the z-figure. Such a trans- formation is a magnification. In the third place we consider w = ze*0. Clearly | w | = | z | and one value of arg w is 0+arg z, and so the w-figure is the z-figure turned about the origin through * Conformed Representation (Cambridge, 1932). 42 FUNCTIONS OF A COMPLEX VARIABLE an angle in the positive sense. Such a transformation is a rotation. Finally, consider w = 1/z. If | z \ = r and arg z = 0, then | w | = 1/r and arg w = 0. Hence, to pass from the t^-figure to the z-figure we invert the former with respect to the origin of the w-plane, with unit radius of inversion, and then construct the image figure in the real axis of the w-plane. The sequence of transformations (4) consists of a combination of the ones just considered ; of these, the only one which affects the shape of the figures is inversion. Since the inverse of a circle is a circle or a straight line (circle with infinite radius), it follows that a Mobius' transformation transforms circles into circles. The transformation inverse to (3) is also a Mobius' transformation z = f (-8)(-a)-y ^ . . (5) yw a Further, if we perform first the transformation (3), then a second Mobius' transformation the result is a third Mobius' transformation _ Az+B { ~ Fz+A where AA-BF = (08 j8y)(a'8'-j8y) ^ 0. Since the right-hand side of (3) is a regular function of z, except when z = 8/y, Mobius' transformations are conformal. If we write the equation of a circle (1) in the form P=0. . . (6) CONFORMAL REPRESENTATION 43 where d and ( are real, and then substitute yw> a yw a in (6) we get an expression of the form Dww+Ew+Ew+F = . . . (7) where D and are both real, and it is easy to verify that the coefficients of w and w are conjugate complex numbers. Hence (7) represents a circle in the u?-plane, since it is of the same form as (1). 14. Geometrical Inversion From what we have just seen it would be natural to expect that there would be an intimate relation between Mobius' transformations and geometrical inversion. Let S be a circle of centre K and radius r in the z-plane. Then two points P and P lf collinear with K, such that KP . KP l = r 2 , are called inverse points with respect to the circle 8, and it is known from geometry that any circle passing through P and P l is orthogonal to S. In the case of a straight line 8, P and P l are inverse points with respect to s, if P! is the image of P in 8. If P, P l9 and K are the points z, Zi, and k we have \(z^k)(z^k)\ = r 2 , arg fo-fc) = arg (z-&), . (8) the second equation expressing the collinearity of the points K y P y P v The two equations (8) are satisfied, if, and only if, (z l -k)(z-Jc) = r 2 . . . . (9) If 8 is the circle dzz+Bz+Bz+ Q = 0, . . . (10) 44 FUNCTIONS OF A COMPLEX VARIABLE which may be written we see that (10) is a circle with centre B/^ and radius BB- Hence equation (9) becomes BB-AQ which on simplification is dz l z+Bz l +Bz+(S=0. . . . (11) We thus get the relation between z and its inverse z l from the equation of S by substituting z l for z and leaving z unchanged. On solving (11), the transformation is We now prove the theorem : The bilinear transformation transforms two points which are inverse with respect to a circle into two points which are inverse with respect to the transformed circle. If z and z 1 are inverse with respect to the circle (10) then (11) is satisfied. Make the transformation (3) and let w and w l be the transformed points. We have . z = - , z = n: r l a ywa and, on substituting these values in (11) we get an expression = where the coefficients D, E, $, F are the same as those of (7) ; in fact we get (7) with w replaced by w v But this is the condition that w and w l are inverse points with CONFORMAL REPRESENTATION 45 respect to the transformed circle (7). The theorem is therefore proved. The inversion (12) can be written as a succession of two transformations -w-(S t0 = Z, Zi=* -77-. 1 Aw+B The first is a reflection in the real axis and the second is a Mobius* transformation. The first preserves the angles but reverses their signs ; the second is conformal. Hence inversion is an isogonal, but not conformal, transformation. Since inversion is a one-one isogonal, but not conformal, transformation, it is clear that the result of two, or of an even number of inversions, is a one-one conformal trans- formation, since both the magnitude and sign of the angles is preserved. In other words, the successive performance of an even number of inversions is equivalent to a Mobius* transformation . 15. The Critical Points If the z-plane is closed by the addition of the point z = oo, then (3) and (5) show that every Mobius' trans- formation is a one-one transformation of the closed z-plane into itself. If y ^ the point w = a/y corresponds to z = oo and w = oo to z = S/y ; but, if y = 0, the points z = oo, w = oo correspond to each other. Since, from (4), dw aS /?y 'dz^ (yz+S) 2 ' the only critical points of the transformation are z = oo and z = S/y. These two critical points cease to be exceptional if we extend the definition of conformal representation in the following manner. A function w = /(z) is said to transform the neighbourhood of a point z conformally into a neigh- bourhood of w = oo, if the function i = l//(z) transforms 46 FUNCTIONS OF A COMPLEX VARIABLE the neighbourhood of z conformally into a neighbourhood of t = 0. Also w = f(z) is said to transform the neighbour- hood of z = oo conformally into a neighbourhood of W Q if w = <() =/(!/) transforms the neighbourhood of = conformally into a neighbourhood of w . In this definition w may have the value oo. With these extensions of the definitions we may now say that every Mobius 9 transformation gives a one-one conformal representation of the whole closed z-plane on the whole dosed w-plane. In other words, the mapping is biuniform for the complete planes of w and z. 16. Coaxal Circles Let a, b y z be the affixes of the three points A, B, P of the z-plane. Then 2 ft * arg = APB, 6 z a if the principal value of the argument be chosen. Let A and B be fixed and P a variable point. FIG. 3,. If the two circles in fig. 3 are equal, z 1? z 2 , z 8 are the affixes of the points P lt P 2 , P 8 and APB = 0, we see that Z 2 * /i 2 i ft * z*~~b arg - - = 7T0 , arg = , arg z a a z, a z 8 a CONFORMAL REPRESENTATION 47 The locus defined by the equation z 6 arg = 0, . . . (1) z a when is a constant, is the arc APB. By writing 0, ^0, 7T+0 for we obtain the arcs AP+B, AP^B 9 AP 3 B respectively. The system of equations obtained by varying from 77 to TT represents the system of circles which can be drawn through the points A, B. It should be observed that each circle must be divided into two parts, to each of which correspond different values of 0. Let T be the point at which the tangent to the circle APB at P meets AB. Then the triangles TPA, TBP are similar and PB BT TP *' Hence TAJTB = k 2 and so T is a fixed point for all positions of P which satisfy -k (2) z-b ~ ' * * * ( ' where k is a constant. Also TP 2 = Tu4 . TB and so is constant. Hence the locus of P is a circle whose centre isT. The system of equations obtained by varying k represents a system of circles. The system given by (1) is a system of coaxal circles of the common point kind, and that given by (2) a system of the limiting point kind, with A and B as the limiting points of the system. If k-+ oo or if k-* then the circle becomes a point circle at A or B. All the circles of one system intersect all the circles of the other system orthogonally. The above important result is of frequent application in problems involving bilinear transformations. It may be used to prove that the bilinear transformation transforms circles into circles. 48 FUNCTIONS OF A COMPLEX VARIABLE Suppose that the circle in the t0-plane is w-X W[JL 1 we substitute for w in terms of z from the bilinear relation w = (3) we obtain z-A' where A' = - a Ay ' ,,__-?.,_ a -Ay k, and so the locus in the z-plane is also a circle. We may write (3) in the form a z+B/a *i\ _ _ _ _ y z+3/y ' and since z+8/y represents a circle in the z-plane, the circle in the w-plane corresponding to it is plainly By taking special values of a, j8, y, S and k the boundaries in the u?-plane corresponding to given boundaries in the z-plane are easily determined. For example, let 0/a = t", 8/y *= * and k = 1 : since the locus \(zi)/(z+i)\ =* 1 is plainly the real axis in the z-plane, this axis corresponds to the circle | w \ = | a/y | in the t^-plane : this will be the unit circle if, in addition, | a | = | y |. CONFORMAL REPRESENTATION 49 17. Invariance ol the Gross-Ratio Let z v z 2 , z a , z 4 be any four points of the z-plane and let w lf u> 2 , w s , u> 4 be the points which correspond to them by the Mobius* transformation d) If we suppose that all the numbers z r , w r are finite, we have '~' ~ y Zf +8 y 2 .+8 ~ (y and hence it follows that (u>i w> 4 )(w 8 w 2 ) (! z 4 )(z 3 (2 1 -Z 2 )(2 3 -2 4 ) (2) The right-hand side of (2) is the cross-ratio of the four points z 1? z 2 , z s , z 4 , and so we have the result that the cross-ratio is invariant for the transformation (1). If equation (2) be suitably modified, it is still true if any one of the numbers z r or one of the numbers w f is infinite. For example, let z 2 = oo and w l = oo, then tt>3-w 2 _ z 1 z i (o) Now suppose that z f , w f (r = 1, 2, 3) be two sets each containing three unequal complex numbers. Suppose first that these six numbers are all finite. Then the equation - -~ 2 ) ' when solved for w leads to a Mobius' transformation which transforms each point z r into the corresponding point w r . The determinant of the transformation has the value -u?^^ 50 FUNCTIONS OF A COMPLEX VARIABLE It is also clear that (4) is the only Mobius' transformation which does so. The result still holds, if (4) be suitably modified, when one of the numbers z r or w r is infinite. The equation (4) above may be used to find the particular transformations which transform one given circle into another given circle or straight line. A circle is uniquely determined by three points on its circumference and so we have only to give special values to each of the tliree sets z r , w r (r = 1, 2, 3) and substitute them in (4). Example. Let z t = 1, z a = i, z 3 = 1 and w l = 0, u>t = 1, w 3 = oo then we get, after substitution in (4), which transforms the circle | z \ = 1 into the real axis of the w-plane and the interior of the circle | z \ < 1 into the upper half of the u>-plane. The easiest way to prove this is as follows. Equation (5) is equivalent to w i 2 ; _ . w+i' The boundary | z \ = 1 corresponds to | w i \ = | w+i |, which is the real axis of the w-plane, since it is the locus of points equidistant from w = i. Since the centre z = of the circle corresponds to the point w = i, in the upper half of the w-plane, the interior of the circle | z \ = 1 corresponds to the upper half of the Similarly, since w = i corresponds to z = oo, the outside of the circle | z \ = 1 corresponds to the lower half of the K;-plane. It may be observed that although this use of the invariance of the cross-ratio will always determine the Mobius' transformation which transforms any given circle into any other given circle (or straight line), it is not necessarily the easiest way of doing so. CONFORMAL REPRESENTATION 51 Thus, in the previous example, since z = 1 and 1 correspond to w =* and oo, the transformation must take the form fc(z-l) ITT' W =3 t corresponds to w = 1, we can determine k : thus fc(t-l) Since z from which it readily follows that k =* i, and we obtain (5) above. 18. Some special Mdbius' Transformations I. Let us consider the problem of finding all the Mobius* transformations which transform the half -plane I(z)^0 into the unit circle \w\ ^1. We observe first that to points z, z symmetrical with respect to the real axis correspond points w, l/w inverse with respect to the unit circle in the w-plane. (See 14.) In particular, the origin and the point at infinity in the w?-plane correspond to conjugate values of z. Let the required transformation be w = yz+8" Clearly y^O, or the points at infinity in the two planes would correspond. Since w = 0, w = oo correspond to z = j8/a, z = S/y we may write j8/a = a, 8/y = a and a 2 a w = -- 1 y z a The point z = must correspond to a point on the circle | w | = 1, so that a y a a a 52 FUNCTIONS OF A COMPLEX VARIABLE hence we may write a = ye*0, where is real, and obtain w = za (1) Since z = a gives to = 0, a must be a point of the upper half-plane, in other words, Ia>0. With this condition, (1) is the transformation required. II. To find all the Mobius* transformations which transform the unit circle \ z (^ 1 into the unit circle \w\^l. Let w = yz+S* In this case w = and w = oo must correspond to inverse points z = a, z = I/a, where | a |<1. Hence j8/a = a, I/a, and so aa za w = - y 2 I/a y dz 1* The point z = 1 corresponds to a point on | u; | = 1, and so aa 1 a aa y = 1. It follows that, if is real, aa = ye*0 and so za . w - i v This is the desired transformation ; for, if z = e*V, a then If z = re*V, where r<l, then r 2 ~2r6 cos (- -r cos ~ CONFORMAL REPRESENTATION 53 hence | w \<1 ; in other words, the interiors of the circles correspond. The identical transformation w = z is a special case of the above : if the point 2 = corresponds to w = 0, then a = and the transformation reduces to w = zeiO. If, in addition, dw/dz = 1 when z = we get w = z. III. The reader should find it easy to verify that the transformation w== p(z-a) maps the circle \ z \ = p on the unit circle \ w \ = 1. If | a |<p it maps | z \<p on | w |<1 and | z \>p on | w If | a | >/> it maps | z \ >p on | w |< 1 and | z \<p on | w \ IV. Representation of the space bounded by three circular arcs on a rectilinear triangle. Consider three circles in the z-plane intersecting at the point 2 = a. B' (w-o) z-plane Fio. 4. w-plane The angles of the curvilinear triangle BCD of fig. 4 are such that a+jS-f y = TT. Consider the transformation 7 w = K a /I (1 x 54 FUNCTIONS OF A COMPLEX VARIABLE The point A(z = a) corresponds to w = oo and B(z = 6) corresponds to w = 0. Equation (1) may be written . k(a-b) wk = , za or w 9 = A/2', where w 9 = wk, z f = za and A = k(a b). Since the changes of variable from w to w' and from z to z' are mere translations, (1) is a pure inversion and reflexion. Since z = a corresponds to w = oo and each of the three circular arcs BC, CD, DB passes through A they correspond to three straight lines in the u;-plane. The two arcs BC, BD which pass through B correspond to two straight lines passing through w = 0, and the arc CD to a straight line which does not pass through w = 0. It readily follows from (1) that the shaded areas of fig. 4 correspond. With the usual convention of sign, we regard a motion round a closed simple contour, such as a circle, in the clockwise sense as positive for the area outside and negative for the area inside the contour. If, by any conformal transformation, three points A, B, C on a closed contour in the z-plane correspond to the three points A', B', C' in the w- plane lying on the corresponding closed contour, then the interiors correspond if the points A', B', C 9 occur in the same (counter-clockwise) order as the points A, B, C. We can see in this way that the shaded areas in fig. 4 correspond. It also follows that the curvilinear triangle formed by the arcs AC, CD, DA in the z-plane corresponds to the portion of the u>-plane A'C'D'A' where A 9 is the point at infinity. EXAMPLES II 1. (i) Prove that, if u = a? 1 y 1 , v = y/(#*+y f ), both u and v satisfy Laplace's equation, but that u+iv is not a regular function of z. CONFORMAL REPRESENTATION 55 (ii) Show that the families of curves u = const, v = const, cut orthogonally if u = x*/y, v = # 2 +2?/ 2 but that the transformation represented by u+iv is not conformal. 2. Prove that, if w = x+iby/a, 0<a<6, the inside of the circle x* +y* = a 1 corresponds to the inside of an ellipse in the to-plane, but that the transformation is not conformal. 3. Prove that, for the transformation w* = (z a)(z /J), the critical points are z = o, z = jS, z = J(a + j3), w = 0, w = $i(a-p). Show also that the condition that z = oo is not a critical point of the transformation w = /(z) is that lim z 2 /'(z) must Jh>00 be finite and not zero. 4. If, by the inversion transformation x = k*(/p* 9 y = fcV P J , * = 2 //> 2 where r P = k *> f2 = z 2 +</ 2 +z*> p* = ^-(-^-[-^ ^ e twice-differentiable function V(x, y, z) becomes V^f, TJ, ) prove that if (d a /df 2 + a 2 /^ 2 + av^D FI=O, then (d*/dx* + B*ldy* + B*/dz 2 ) (V/r) = 0. 5. If w = cosh z, prove that the area of the region of the u;-plane which corresponds to the rectangle bounded by the lines x = 0, x = 2, y = 0, y = JTT is (TT sinh 4 8)/16. 6. If a is real and O<C<TT, find the area of the domain in the u;-plane which corresponds by the transformation w = e* to the rectangle a c^.x^.a+c 9 c<t/<c. Find the ratio of the areas of the two corresponding domains and prove that the ratio ->- e 20 as c -> 0. 7. Show that, if the function w =/(z), regular in | z \< R, maps the circle | z \ = r<R on a rectifiable curve G in the u>-plane, then the length of C is given by r I J \f'(reie)\rd0. o Show that the length of the curve into which the semi- circular arc | z | = 1, Jn"^argz<i7r is transformed by w = 4/(l+z) 2 is 2V2+2 log(l + \/2). (See 22, equation (4)). 8. Find the Mobius' transformations which make the sets of points in the z-plane (i) a, 6, c, (ii) 2, 1 -ft, to the points 0, 1, oo of the u^ane. In case (ii) 56 FUNCTIONS OF A COMPLEX VARIABLE sketches the domains of the u?-plane and 2 -plane which correspond. 9. Find a M6bius* transformation which maps the circle 1*1^1 on | w 1 | ^ 1 and makes the points z = , 1 correspond to w = , respectively. Is the transformation uniquely determined by the data ? 10. Find the transformation which maps the outside of the circle | z \ = 1 on the half -plane Ru;>0, so that the points 2 = 1, i, I correspond to w = i, 0, i respectively. What corresponds in the w- plane to (i) the lines arg z = const., | z | ^ 1, (ii) the concentric circles | z \ = r, (r>l) ? 11. Prove that w = (l+iz)/(i+z) maps the part of the real axis between 2 = 1 and z = 1 on a semicircle in the ttf-plane. Find all the figures that can be obtained from the originally selected part of the axis of x by successive applications of this transformation. 12. Find what regions of the w-plane correspond by the transformation w = (z i)/(z+i) to (i) the interior of a circle of centre z = t, (ii) the region t/>0, x>0 9 \z+i\ < 2. Illustrate by diagrams. Show that the magnification is constant along any circle with z = i as centre. 13. Let GI, | zz l | =5 T! and C 8 , | z z % \ = r 8 be two non -concentric circles in the 2-plane, C l lying entirely within C f . Show that, if z = a, 2 = 6 are the limiting points of the system of coaxal circles determined by C l and <7 a , then w = k(z 6)/(2 a) transforms C l and (7, into concentric circles in the w -plane with centres at w = 0. If the radii of these concentric circles are p l and p 2 , show that, although there is an infinite number of such representations, p t : p f is a constant. 14. Prove that, if w = (oz + 0)/(y2 + 8) and a8 0y = 1, then the linear and superficial magnifications are | y2 + S j" 1 , Show that the circle | y2 + 8 | = 1 (y ^ 0) is the complete locus of points in the neighbourhood of which lengths and areas are unaltered by the transformation. Prove that lengths and areas within this circle are increased and lengths and areas outside this circle are decreased in magnitude by the transformation. (This circle is called the isometric circle.) CHAPTER III SOME SPECIAL TRANSFORMATIONS 19. Introduction The fundamental problem in the theory of conformal mapping is concerned with the possibility of transforming conforrnally a given domain D of the z-plane into any given domain D' of the w-plane. It is sufficient to consider whether it is possible to map conforrnally any given domain on the interior of a circle. For if = f(z) maps D on | C|<1 and w = F(Q maps D' on | |<1, then w = F{f(z)} provides a conformal transformation of D into D'. The fundamental existence theorem of Riemann states that any region with a suitable boundary can be conforrnally represented on a circle by a biuniform transformation. Rigor- ous proofs of this existence theorem are long and difficult, and it is beyond our scope to discuss the question here. In the applications of conformal transformation to practical problems, the problem to be solved is as follows : given two domains D and D' with specified boundaries, find the function w = f(z) which will transform D into D' so that the given boundaries correspond. Although, by Riemann 's existence theorem, we can infer the existence of the regular function /(z), the theorem does not assist us to find the particular function f(z) for each problem whose solution is desired. We have seen that when the two domains D and D' are bounded by circles, it is fairly easy to find the Mobius' transformation which maps D biuniformly on Z)'. Since for any arbitrary boundary curves there is no general method of finding the appropriate 58 FUNCTIONS OF A COMPLEX VARIABLE regular function f(z), it is important to know the types of domain which correspond to each other when f(z) is one of the elementary functions or a combination of several such functions. In this chapter we discuss some of the most useful transformations which can be effected by elementary functions. The reader, who is mainly interested in the application of these transformations to practical problems, will find the special transformations discussed here of great value, but he must refer to other treatises for the details of the practical problems to which they can be applied. Many useful transformations are obtained by combining several simple transformations. For example, the transformation seems at first sight somewhat complicated, but on examination it is seen to be a combination of the successive simple trans- formations, Z = z 8 , == - - , t = *, w == . 1 6 t-\-l It can be shown that (1) maps the circular sector | z j< 1, 0<arg z< JTT, conformally on the unit circle * | w |< 1. 20. The Transformations w = z n Let w = u+iv = pety, z = x+iy = re^, then it follows at once that p = r n , < = n6, so that u+iv = r n (cos nO+i sin n0). From the equations u = r n cos ri0, v = r n sin n0, . . (1) * CJompare 24 IV, equation (8), with a }. SOME SPECIAL TRANSFORMATIONS 59 either or r may be eliminated, giving f . . (2) or tan n0 = - . . . (3) u ^ ' Equation (2) shows that the circles r = c of the z-plane and the circles p = c n correspond, and in particular, that points on the circle r = | z \ = 1 are transformed into points in the to-plane at unit distance from the origin. The lines = const., radiating from the origin of the z-plane, are transformed into similar radial lines <j> = const. It should be noticed, however, that the line whose slope is in the z-plane is transformed into the line whose slope is n0 in the w-plane. Since z = is a critical point of the transformation, the conformal property does not hold at this point. In the simple case w = z 2 , the angle between two radial lines in the z-plane is doubled in the w-plane. The case w = z 2 is typical, and we shall now consider it in greater detail. We consider first the important difference between the transformation w = z 2 and the Mobius' transformations discussed in the preceding chapter. In the latter, points of the z-plane and of the t0-plane were in one-one correspondence. For w = z 2 , to each point z there corresponds one and only one point W Q = z^, but to a point w Q there correspond two values of z, z= JV^ol* z = \\/w \. If we wish to preserve the one-one corres- pondence between the two planes, we may either consider the w-plane as slit along the real axis from the origin to infinity, or else construct the Riemann surface in the t0-plane corresponding to the two-valued function of w defined by w = z 2 . The method of constructing the Riemann surface was described in 9. If we use the cut w-plane, then the upper half of the -plane corresponds to the whole cut u?-plane. There is 60 FUNCTIONS OF A COMPLEX VARIABLE a one-one correspondence between points of the upper half of the z-plane and points of the whole u?-plane, and a one-one correspondence between points of the lower half of the z-plane with points of the whole t0-plane ; but if we choose one of the two branches of w = z 2 , say w v the cut plane effectively prevents our changing over, without knowing it, to the other branch w 2 . The positive real z-axis corresponds to the upper edge and the negative real z-axis to the lower edge of the cut along the positive real axis in the u;-plane. If we use the two-sheeted Riemann surface in the w?-plane, the sheet P l corresponds to the upper half of the z-plane for the branch w l9 and the sheet P 2 corresponds to the lower half of the z-plane for the branch w 2 . Thus there is a one-one correspondence between the whole z-plane and the two-sheeted Riemann surface in the u>-plane. For w = z n , where n is a positive integer, a wedge of the z-plane of angle 2n/n corresponds to the whole of the t0-plane. If we divide up the z-plane into n such wedges, each of these corresponds to one of the n sheets of the n-sheeted Riemann surface in the w-plane. If, for w = z 2 , we cut the u?-plane along the negative real axis, then the sheet P l of the Riemann surface corresponds to the half-plane Rz^O, and the sheet P 2 to the half-plane Rz<0. 21. Further Consideration ol w = z 2 From the equations w = u-}-iv = (x-\-iy) 2 = x 2 - y z +2ixy, we have u = x 2 i/ 2 , v = 2xy. . . (I) By regarding u and v as curvilinear coordinates of points in the z-plane, the transformation w = z 2 can be examined from a knowledge of the curves in the z-plane which SOME SPECIAL TRANSFORMATIONS 61 correspond to constant values of u and v. This method is frequently used in applying the theory of conformal transformation to practical problems. Equations (1) show that the curves u = const., v = const, in the z-plane are two orthogonal families of rectangular hyperbolas. The reader will easily verify that the shaded area in the z-plane of fig. 5 between two hyperbolas 2xy = v lt z-plane FIG. 5. u;-plane 2xy = v 2 corresponds to the infinite strip of the t^-plane shaded in the figure. Hence w = z a maps the region between two hyperbolas on a parallel strip. If B l is at infinity, the point B\ is also at infinity, and the interior of the hyperbola ABC is transformed into the part of the upper half-plane above the line A' B' C'. We also observe that the transformation w = z* makes circles \ za \ = c, (a, c real) 9 in the z-plane correspond to limaqons in the w-plane. Consider the circle 2 a = ce*"*, ... (2) then t/* -a 2 + c 2 == 2c(c cos 9 f 62 FUNCTIONS OF A COMPLEX VARIABLE Hence, on writing w a 2 -j-c a = Be^ 9 so that the pole in the w-plane is at w = a 2 c 2 , the polar equation of the curve into which (2) is transformed is the Iiina9on B = 2oc+2c 2 cos 0. When a = c the Iima9on becomes a cardioid. This is the case if the circle (2) touches Oy at the origin, 22. The Transformation w = y'* Prom the equations u*v* = x, 2uv = y, . * (1) we get . . (2) By means of the first of the equations (2), to the straight lines u = const, correspond parabolas with vertex at x = w a and focus at the origin of the z-plane. To the orthogonal system of straight lines v = const., we see, by the second of the equations (2), there corresponds another system of confocal parabolas with vertex at x = v 2 . Consider the particular parabola of the first system corresponding to the value u = 1, y 2 = 4(l-*). ... (3) Its transform in the w-plane is the line through the point w = 1 parallel to the t;-axis. The points, A, B, C in fig. 6 correspond, in that order, to the points A', B', C' . The reader can easily verify that the shaded areas correspond, the two parabolas drawn corresponding to values u = 1, and u = ^ (>1). If w -> oo, the region developed in the 2-plane is the area outside the parabola (3), which accordingly corresponds to that part of the t#-plane to the right of the line u = 1. If the parameter u tends to zero, the parabola y 2 = 4u 2 (u 2 x) narrows down until it becomes a slit along the negative real axis OX 19 which is a branch-line. SOME SPECIAL TRANSFORMATIONS 63 Hence the portion of the w-plane between the line u = 1 and the line u = corresponds to the portion of the z-plane between the parabola ABC, y* = 4(1 x), and the cut along the negative real axis OX l from the origin to --00. 2-plane FIG. 6. c f plane Hence we see that the portion of the w-plane corresponds to the whole z-plane cut along the negative real axis from to -co. The simple w-plane is associated in a one-one corre- spondence with a two-sheeted Riemann surface covering the z-plane. The two sheets of the Riemann surface would be connected along the edges of the cuts along the negative real axis of the z-plane in the usual way. The line u = 1 plainly corresponds to the same parabola y a = 4(l x) as does the line u = 1. Hence the portion of the w-plane to the left of the line u =* 1 corresponds to the region outside the parabola ABC which lies on the second sheet of the Riemann surface. If we combine w = y'z with a Mobius' transformation by writing = (2/w)-~ 1 we see that the transformation C---1 (4) fc / * v*; transforms the region outside the parabola (3) into the interior of the unit circle in the {-plane. The points z = 1, z = 4, 64 FUNCTIONS OF A COMPLEX VARIABLE z = oo correspond to the points = 1, = 0, {=1. The focus of the parabola (3), z = 0, lies outside the region of the z-plane which is under consideration, and it corresponds to the point = oo outside the unit circle Kl-i. The reader should observe, however, that the preceding transformation cannot be used to represent the inside of the parabola ABC on the inside of the unit circle. The transformation w = \/z just considered, illustrates an important point in the use of many-valued functions for solving problems in applied mathematics. The transformation w = ^z could be used to deal with a potential problem in which the field was the region outside the parabola ABC of fig. 6, but it could not be used for a problem in which the field was the space inside this parabola, since two points close to each other, one on each edge of the cut along the branch-line OX l9 will transform into two points on the axis of v, one in the upper and the other in the lower half -plane. Since these points are not close together in the t-0-plane, they would correspond to different potentials. It is important to realise that we cannot solve potential problems by using transformations which require a branch -line to be introduced into that part of the plane which represents the field. 23. The Transformation w = tan 2 (j7r\/z) We have just seen that w = (2/\/z) l cannot be used to map the region inside the parabola t/ 2 = 4(1 x) on the unit circle |t0|^l. We now consider a transformation which enables us to do this. The transformation can be considered as a combination of the three transformations w = where w = u+iv, = +^'77, t = a-ftY, z = x+iy. The first transformation can be written Icos "*" l+cos* SOME SPECIAL TRANSFORMATIONS 65 If we consider the infinite strip between the lines = 0, = J^TT of the -plane, we see that, by writing f = far -\-irj, cos = i sinhrj and | w \ = 1. Thus, as y goes from oo to oo along the line g = far, w describes the unit- circle once. By writing f = irj , cos = cosh rj and w; is real. Thus as 77 goes from +00 to 0, w goes from 1 to ; and as 77 goes from to oo, w retraces its path from to 1. Thus the strip = 0, = \n corresponds to the cut-circle as illustrated in fig. 7. It is easy to verify to-plane f-plane J-plane z -plane FIG. 7. that the interiors correspond. The strip in the 2-plane is plainly that between the lines a = and a = 1. As we have already seen, t = \/z transforms the strip in the -plane into the region inside the parabola ABC, f/ 2 = 4(l x), with a cut from the origin to infinity along the negative real axis. In fact, as or -> the parabola y 2 = 4a 2 (or 2 x) becomes a very narrow parabola which is the slit illustrated in the z-plane in fig. 7. The transformation w = tan 2 (7r\/z) represents the region inside the parabola ABO on the inside of the unit circle | w \ = 1 in a one-one correspondence, for the real axis of the w-plane between 1 and corresponds to the real axis of the z-plane between oo and 0. The cuts in the z-plane and w-plane are not needed for the direct trans- formation from the w>-plane to the z-plane, but they are needed for the subsidiary transformations used in order to show how the boundaries of the various regions correspond. 66 FUNCTIONS OF A COMPLEX VARIABLE Since dw/dz = 77 tan(}rr\/z) sec 2 ( j7r\/ 2 )/4 \/ z > w ^ich tends to a finite non-zero limit as z-> 0, the points z = and u> = are not critical points of the transformation, and so the representation is conformal as well as one-one. 24. Combinations of w = z a with Mobius' Trans- formations I. Semicircle on half-plane or circle. Consider the transformation / z _ j c \ 2 ' (creal) - (1) This is clearly a combination of w = 2 and = (z ic)/(z+ic). The second of these may be written * . +1 for which it is clear that the circle | z | = c corresponds to the imaginary axis of the -plane |-fl| = | 1|. The boundary of the semicircle in the z-plane A DCS A plainly corresponds to A'D'C'B'A' in the -plane, C' being the point = -oo. The sense of description of the two boundaries shows that the shaded areas correspond. Now consider w = a : if w = pety, = re*0, we have The shaded domain of the -plane corresponds to 7r<0<37r/2 and so the domain of the w-plane corresponding to this is 27r<<<37r, which is of course the same as r, or the upper half of the w-plane. * The use of the results of 16 is frequently simpler than the procedure of splitting up the transformation into its real and imaginary parts. SOME SPECIAL TRANSFORMATIONS 67 Hence the interior of the shaded semicircle of fig. 8 corresponds to the upper half of the w-plane. z-plane FIG. 8. It is easy to verify that the upper half of the w-plane corresponds to the interior of the respective semicircles BAEDB, AECDA, ECBDE by the transformations {z+c\* w = I 1 , w _ w - Also, by combining (1) with the transformation the transformation iw t == i+w' t = \z+icj i + (*!=*]* . z a 2 a C 2 2CZ conformally represents the interior of the z-semicircle ABCDA on the interior of the unit circle 1 1 1 = 1. II. Wedge or sector on half -plane. By the transformation w = 2 1 / , (2) 68 FUNCTIONS OF A COMPLEX VARIABLE the area bounded by the infinite wedge of angle na with its vertex at z = and one arm of the angle along the positive x-axis is transformed into the upper half of the w-plane. The reader will find this quite easy to verify. The sector cut off from this wedge by an arc of the unit- circle | z | = 1 is transformed by (2) into the unit semicircle in the upper half of the w-plane. This is also easy to verify. III. Circular crescent or semicircle on half -plane. We readily see, from 18, IV, that the circular crescent with its points at z = a and z = b and whose angle is TTOL can be transformed into the wedge mentioned in II above by z ~ a if the constant k be suitably chosen. Hence the crescent can be transformed into the w-half-plane by -a\l/a =} i) () A semicircle may be regarded as a particular case of a crescent in which a = J. The semicircle of radius unity and centre 2 = lying in the upper half-plane is trans- formed into the first quadrant of the -plane by The quarter-plane becomes a half-plane by * w = 2 and so the semicircle is transformed into the upper half of the w-plane by =<=()' < 6) * See I above. SOME SPECIAL TRANSFORMATIONS 69 IV. Sector on unit circle. Consider the sector in the z-plane, shaded in fig. 9. Let us find the transformation which represents this sector on a unit circle. o j Fia. 9. By means of (2) the sector is transformed into the unit- semicircle in the upper half of the w-plane. By means of (5) we see that this unit semicircle is transformed into the upper half of the J-plane by (6) Again, the upper half of the -plane is transformed into the interior of the unit circle in the -plane, | f | = 1, by .... (7) l-fz 1/a \ 2 and on combining these, the transformation which represents the shaded area of fig. 9 on the unit circle in the {-plane is * V. By combining w = \/z with a Mobius' transforma- tion we find in a similar way the transformation which represents the z-plane, cut from to oo along the positive real axis on the unit circle \ | < 1 in the form When a = J, this transformation is the same as (1) of 19. 70 FUNCTIONS OF A COMPLEX VARIABLE VI. Transformations of the cut-plane. Consider the two transformations z a za W = r , W = r zb zb where a and b are real and a> 6. By means of the first of these, the z-plane, cut along the real axis from z = a to +00 and from z = b to oo, is transformed into the usplane cut from w = to w = oo, the cut passing through the point w = 1 which corresponds to z = oo. By means of the second, the 2-plane cut from z = a to z = 6 is transformed into the u;-plane cut along the positive real axis from to oo. The cut in this case does not pass through w = 1, the point corresponding to z = oo. 25. Exponential and Logarithmic Transformations Most of the transformations so far considered have been Mobius' transformations, w = z a and combinations of these two types. We now observe that the relation t* = e . . . . (1) gives rise to two important special transformations. If we use rectangular coordinates x, y and polar co- ordinates />, <f> in the to-plane we get p = e * , <f> = y. The horizontal strip of the 2-plane bounded by the lines y = y l and y = y 2 where | y^y^ |<27r is transformed into a wedge-shaped region of the t^-plane, the angle of the wedge being a = |< 2 ~~^i I ^ I V^^Vi I- The repre- sentation is conformal throughout the interior of these regions since dw/dz is never zero. In particular, if y l = 0, y 2 = TT, so that | y^y\ \ = TT, the wedge becomes a half- plane. The semi-infinite strip oo<#<0, 0<y<7r is readily seen to correspond to unit semicircle in the upper half of the u>-plane. If | yiy^ |>2rr, the wedge obtained covers part of the SOME SPECIAL TRANSFORMATIONS 71 w-plane multiply. We may in this case make use of the cut w-plane. If y l = , y a = 27r, the strip of width 2ir in the z-plane corresponds to the w-plane cut along the positive real axis. When | s^ 1/ 2 \ is an integral multiple of 277 the strip is transformed into a Riemann surface. Each strip of the z-plane of breadth 2rr corresponds to one sheet of the oo-sheeted Riemann surface. A second special transformation is obtained from (1) by considering an arbitrary vertical strip bounded by the lines x = x l , x = x 2 , (x^x^. This strip is represented on a Riemann surface which covers the annulus between the concentric circles \w\=p l ,\w\=p 2 an infinite number of times. If we keep # 2 constant and let x l -^~ oo, the strip x l <x<x 2 becomes in the limit the portion of the z-plane to the left of the line x = # 2 , and we obtain in the w-plane a Riemann surface which covers the circle | w |</> 2 except at the point w = 0, where it has a logarithmic branch-point. The inverse function w = Log z . . . (2) gives, on interchanging the z-plane and u;-plane, exactly the same transformations as (1). It should be remembered that although Log z is an infinitely many- valued function of z, e* is one-valued. Since z a = e a Log * the transformation w = z a may be regarded as a combination of the two transformations w = ef , = a Log z. 26. Transformations involving Confocal Conies Consider the transformation (1) If to = reW, we get 2x = |(a 6)r + ^tH cosl9, 2y = j(a-6)r- a \ sin 0, n FUNCTIONS OF A COMPLEX VARIABLE and so the curves in the z-plane, corresponding to concentric circles in the w-plane having the origin for their centre, are confocal ellipses, the distance between the foci being 2<v/(a 2 6 2 ). The curves in the z-plane corresponding to straight lines through the origin in the u;-plane are the confocal hyperbolas, a result to be expected, since the two families of curves in each plane must cut orthogonally. Clearly there is no loss of generality by taking a = 1, 6 = 0, and so we may consider the transformation 2z = w + - . . . (2) w as typical. Clearly z becomes infinite when w = 0, and since dw ~~" : the points w = 1, at which the derivative vanishes, are critical points of the transformation. We now have 2x = I r -f- ~ I cos , 2y = I r I sin and on eliminating 0, we get the ellipse in the z-plane rr 2 */ 2 = 1 , . . (3) corresponding to each of the two circles | w \ r , | w \ = 1/r. As r->l, the major semi-axis of the ellipse tends to 1, while the minor semi-axis tends to zero. As r-> 0, or as r-> oo, both semi-axes tend to infinity. From this it is plain that the inside and the outside of the unit circle in the w-plane both correspond to the whole z-plane, cut along the real axis from 1 to 1. The unit circle | w \ | = 1 itself corresponds to a very narrow ellipse, which is the cut along the real axis enclosing the critical points 1 and 1. SOME SPECIAL TRANSFORMATIONS 73 On solving equation (2) for w we get w = zA/(z 2 -l) and the inverse function is a two-valued function of z. If we choose the lower sign, the transformation w = z -V(z*-l) ... (4) transforms the area outside the ellipse (3) conformaHy into the inside of the circle \ w \ = r. The lower sign is the correct one to select, since the point w = inside the circle must correspond to the point z = oo ; the other sign of the square root would make the points w = oo, z = oo correspond. The region between two confocal ellipses in the z-plane is transformed into the annulus between two concentric circles in the w-plane. The function (4) gives a transformation of the z-plane, cut along the real axis from 1 to 1, on the interior of the circle \ w \ = 1. If we take the other sign, it is clear that the trans- formation w = gives a transformation of the cut z-plane on the outside of the unit circle \ w \ = 1. The relation (2) is remarkable in that it represents the cut z-plane not only on the interior but also on the exterior of the unit circle | w \ = 1. The ambiguity can be removed from this transformation by replacing the z-plane by a Riemann surface of two sheets, each cut from 1 to 1 and joined crossways along the cut. Then, of course, the interior of the unit circle | w | = 1 corresponds to one sheet and the exterior of the unit circle to the other sheet of this Riemann surface. If r>l, the transformation (2) maps the exterior of the circle | w \ = r, or the interior of the circle | w \ = 1/r, on the exterior of the ellipse (3). It should be observed, however, that the interior of the ellipse cannot be represented on the interior of the unit circle by any of the 74 FUNCTIONS OF A COMPLEX VARIABLE elementary transformations so far employed. We may remark, however, that the upper half of the ellipse (3) is represented by (2) on the upper half of an annular region cut along the real axis : this last area, and hence the semi-ellipse also, can be transformed into a rectangle by a method similar to that described in 25. The transformation which maps the interior of an ellipse on a unit circle involves elliptic functions. 27. The Transformation z = c sin w From the relation z = csin;, (c real) . . (1) we get, on equating real and imaginary parts, x = c sin u cosh v, y = c cos u sinh v, so that, when v is constant, the point z describes the curves 2 2 i /o\ ' * * l ' C 2 cosh 2 t; c 2 sinh 2 i; which, for different values of t>, are confocal ellipses. Consider a rectangle in the w-plane bounded by the lines u = i^j v = A. For all values of u, cos u is positive ; hence when v = A, y is positive and x varies from c cosh A to c cosh A, that is, the half of the ellipse on the positive side of the axis of x is covered. Let u = 77-, then y = and x = c cosh v. Hence as v varies from A through zero to A along the side of the rectangle, x passes from A' to the focus H (see fig. 10) and back from H to A'. When v = A then z describes the half of the ellipse on the negative side of the axis of x. When u = far then y = and x = c cosh v, so that z moves from A to the focus S and back from 8 to A. SOME SPECIAL TRANSFORMATIONS 75 Hence the curve in the z-plane corresponding to the contour of the rectangle in the w-plane is the ellipse with two slits from the extremities of the major axis each to the nearer focus. It is easy to see that the two interiors correspond. FIG. 10. Since sin w = COS(JTT w), the transformation given by z = c cos w can be dealt with in a similar way. The details are left to the reader. The function inverse to (1), w = arc sin (z/c), is an infinitely many- valued function of z. If we use the cut plane, the cuts must be from S to infinity along the positive real axis and from H to infinity along the negative real axis. The Riemann surface of an infinite number of sheets in the z-plane, which would secure unique corres- pondence between every z-point and every w-poini, would have the junctions of its different sheets along the above- mentioned cuts. 76 FUNCTIONS OF A COMPLEX VARIABLE 28. Joukowski's Aerofoil The transformation WKC lzc\* I I 4 t ^ \-*-J UJ+/CC \2+<V is important in the practical problem of mapping an aeroplane-wing profile on a nearly circular curve. If the profile has a sharp point at the trailing edge and we write jg = (2 JC)TT, then /? is the angle between the tangents to the upper and lower parts of the profile at this point. If a circle is drawn through the point c in the 2-plane, so that it just encloses the point z = c and cuts the line joining 2 = c and 2 = c at 2 = c+* where is small, this circle is mapped by (1) on a wing-shaped curve in the w-plane. A special case of (1) when c = 1, K = 2 will now be discussed in detail. In practical problems on the study of the flow of air round an aerofoil, the transformation desired is one which maps the region outside the aerofoil on the region outside a circle or nearly circular curve. The special case of (1) when c = 1, *= 2, transforms a circle in the 2-plane, passing through the point z = 1 and containing the point 2 = 1, into a wing-shaped curve in the w?-plane, known as Joukowski's profile. We readily see that (2) is the same as the trans- formation, already discussed, w = z + - . . . . (3) z If C is a circle in the 2-plane passing through the point 2 as 1, such that the point z = 1 is within (7, the trans- formation (3) maps the outside of C conformally on the SOME SPECIAL TRANSFORMATIONS 77 outside of Joukowski's profile F. The shape of the curve F can easily be obtained from the circle G by making the point z trace out this circle, and adding the vectors z and 1/z. See fig. 11. FIG. 11. We may also consider (2) as a combination of the three transformations 2+1 / 2-1 = *, 10: By the first of these, the circle C is transformed into a circle F in the -plane passing through t == 0. By the second, the circle F is transformed into a cardioid * in the -plane with cusp at = 0. The third transformation then maps the cardioid on the wing-shaped curve F in the w-plane. Since z = 1 corresponds to J = oo, the outside of C is mapped on the interior of F. The interior of the cardioid corresponds to the interior of F. Since = 1 corresponds to w = oo, the outside of F corresponds to the inside of the cardioid, and so to the outside of C. In fig. 11, C is the given circle, C' the circle obtained from C by the transformation l/z and Q is the unit circle. * See 21. 78 FUNCTIONS OF A COMPLEX VARIABLE The critical points of (3) are z = 1 and z = 1, and since the point z = 1 is inside C, the mapping of the outside of C on the outside of F is conformal. If the point z = 1 is outside (7, then the inside of F corresponds to the inside of (7, and the figure corresponding to this case is the same as fig. 11 with the circles G and C' interchanged. 29. Some Important Transformations Tabulated In Table 1 we tabulate for convenience a number of examples of domains in the 2-plane which can be mapped conformally on the interior of the unit circle | w | TABLE 1 Domain in the z-plane Domain in the u?-plane Transformation. 1, Unit circle |f |<1 Unit circle | w |^1 iX *~ W ~ C dz-1 2. Upper half -plane Unit circle | w\^.l \z 3. Infinite strip of finite breadth oo^/^oo, Unit circle | w|^l w ~ l _ .iz w+l ** 4. Area outside the ellipse Unit circle | u>|<l w 6. 6. 7. Area outside the para- bola r cos 2 J0 = 1 Area within same para- bola Semicircle Unit circle | w |^1 Unit circle | u;|^l Unit circle | t0|^l w = tan 2 (i77\/2) w ~~~ i Some useful conformal transformations in which the domain in the to-plane is not a circle are given in Table 2. When the domain in the u>-plane is either the upper half- plane or a semicircle it can of course be transformed into a circle by either 2 or 7 of Table 1. SOME SPECIAL TRANSFORMATIONS 79 i "*" > + J? 1 > ^> > I I II II II * 3 S S _j_ + + S S II S lane Domain S o O < V A Upper half -plan Upper half -plan l S : o V V* C5 w |< cj a o 2 ^ | o 1 A ^ i * .2 II OQ I s plane. e angle 0<0<27r/n icircle | z \ < 1, t/> V ii :A.S-XT; % J39^ - "So-s^-c "K o H I 2 $ 3 : > . 5 O *O t^QO 80 FUNCTIONS OF A COMPLEX VARIABLE It is impossible, in the limited space at our disposal, to discuss all the transformations which are of practical importance. It is important, however, to mention briefly the Schwarz-Christoffel transformation, which has numerous important applications. Let a, 6, c, ... be n points on the real axis in the to-plane such that a<6<c<... ; and let a, jS, y, ... be interior angles of a simple closed polygon of n vertices so that a+j3+y+... = (n-2)7T. Then the transformation of Schwarz-Christoffel is a transformation from the w-plane to the 2-plane defined by dz ?! ft + y , ~ ~ It transforms the real axis in the w-plane into the boundary of a closed polygon in the 2-plane in such a way that the vertices of the polygon correspond to the points a, 6, c, ... and the interior angles of the polygon are a, j8, y, .... When the polygon is simple, the interior is mapped by this transformation on the upper half of the to-plane. The number K is a constant, which may be complex. If we write K = Ae*\ where A and A are real, one vertex of the polygon can be made to correspond to the point at infinity on the to-axis. If a-* oo we can choose (X A to be of the form B(a)~" +l and since, as a-> oo, --i {(w a)/ a}^ -> 1, the transformation becomes dz > P . y_, t - l The reader who desires further information about this important transformation is referred to larger treatises.* * See e.g. Copson, Functions of a Complex Variable (Oxford, 1935), p. 193 seq. SOME SPECIAL TRANSFORMATIONS 81 EXAMPLES III 1. Prove that, by the transformation c tza z+a w - two sets of coaxal circles are transformed into sets of confocal conies. What region of the t0-plane corresponds to the inside of the circle \(z a)/(z+a)| = J ? (z-\~ - -. + Z I the real axis in the z-plane corresponds to a cardioid in the to-plane. Indicate the region of the 3-plane which corres- ponds to the interior of the cardioid. 3. If w = ic cot \z y where c is real, show that the rectangle bounded by x = 0, x = IT, y = 0, y = oo, is confonnally repre- sented on a quarter of the to-plane. Find a transformation =/(z) which maps this infinite rectangle on the semicircle ||<a, >0. 4. If w = tan z, prove that cot 2a?-l = 0, u'+v'-^v coth 2y+l = 0. Hence show that the strip %TT<X<%TT corresponds to the whole to-plane. To obtain a Riemann surface in the to-plane so as to secure unique correspondence between every t0-point and every z-point, show that the to-plane must be cut along the imaginary axis from i to oo and from t to oo. Investigate to = tan z as a combination of iw = (-!)/(+!), = 6 . 6. Find the curves in the z-plane corresponding to | to | = 1 if w = 6. Show that w = 2z/(l z 1 ) maps two of the four domains, into which the circles | z 1 | = V2, | z + l \ = V2 divide the 2-plane, confonnally on | w \ < 1. 7. Prove that, if 3z a 2w + l = 0, the annulus l/\/3<| z \<l is mapped confonnally on the interior of the 82 FUNCTIONS OF A COMPLEX VARIABLE ellipse u*4-4y | = 4 cut along the real axis between its foci. Discuss what corresponds in the t0-plane to the curves (i) | z | = r, (ii) arg z = a. 8. Find the transformation which maps the outside of the ellipse | s2 |+| z+2 \ = 100/7 on the circle | w |<1. 9. Show that w = $(z + l/z) maps the upper half of the circle |z|<l on the upper half of the w-plane. At what points of the z-plane is the linear magnification equal to ? At what points is the rotation equal to 77 ? Prove that the magnification is greater than unity throughout the interior of the semicircle 3 | z | 2 = 1 in the upper half -plane. 10. By considering the successive transformations = (z + l/z), w = l/ a prove that w = 4z 2 /(l+z 2 ) a maps the upper half of the circle | z |< 1 on the w-plane, cut along the positive real axis, so that the points z = 0, 1, i correspond to w = 0, 1, oo respectively. What points of the t0-plane correspond to z = 1 ? 11. Show that by w = e 77 " 2 /* an equiangular spiral in the w -plane corresponds to a straight line in the z -plane. 12. Discuss the transformation showing that the lines u = const., v = const, correspond to sets of confocal conies with foci at z = a, z = j8. 13. Show that 2w = log{( l+z)/( lz)} represents |z|<l on the strip of the w-plane JTT<V< JTT. 14. Show that iw = log{V(^/)~ 1} represents the strip v = 0, v = oo, u = TT, u = TT on the interior of the cardioid r = 2a(l+cos 0) in the z -plane cut along the real axis from the cusp to x = a. 15. Show that, if c is real, -=? z+c conformally transforms the strip v= oo, t>= oo, w = 0, u s=s IT, into the circle | z \ ^ c. 16. Show that, by the relation w 2 = l+e f , the linos x = const, are transformed into a series of confocal lemniscatea (Cassini's ovals) in the w-plane. SOME SPECIAL TRANSFORMATIONS 83 If a>l and Z 2 (a 8 +t0* 1) = aw*, show that the interior of the circle | z \ = 1 is transformed into the interior of the Cassini's oval pp = a, where p and p are the distances of a point from the foci (1, 0) and ( 1, 0). 17. If z = x+iy, prove that the inside of the parabola y* = 4c*(a;+c a ) is mapped on the upper half of the u>-plane by _ u? = t cosh 2c 18. Show that the transformation Z C ~ l-W transforms the inside of the circle | w \ = 1 with two semi- circular indentations, of centres 1 and 1, drawn so as to exclude these points from the circular area and boundary, into the annulus between two circles in the z -plane, of centre the origin and radii ce a , ce~, with a single slit along the real axis. 19. If 0<a< 0<27T, show that w ^ (ze- maps the region a<argz</J on the K;-plane cut along the positive real axis. Hence find the transformation w =/(z) which maps the circular sector a<argz<0, |2J<1, on the circle | w |<1. 20. Use the successive transformations C = (z+iM s = e*", t = - - , r = t\ w = -., 18 r+i to form the single transformation w =/(z) which maps the strip <#<$, J/5^0 of the z-plane on | w |^1. 21. Use the transformations = y% ^ = sin TT, w = - , to show that sin irr\/ 2 1 W s - sin maps the inside of the parabola r = 2/(l -fcos 0) in the z-plane, 84 FUNCTIONS OF A COMPLEX VARIABLE out from the focus (z = 0) to the point z = oo, on the unit circle in the t0-plane cut from w = to w = 1. 22. Find the equations of the curves in the s-plane which correspond to constant values of u and v if z = w+e w . What corresponds to the lines v = 0, v = ir ? Sketch some of the curves v = const, for values of v between TT and IT. 23. Show that the transformation w/a = i sinh \(z ij8)/cosh (z+if)) transforms one of the regions bounded by the orthogonal circles | w \ = a and | w a cosec ft \ = a cot ft into the infinite strip 0<t/<j7T. 24. If w = tanhz, show that the lines x = const., y = const., correspond to coaxal circles in the w-plane. Prove that this transformation maps the strip conformally on the upper-half of the w-plane. 25. Prove that, if 0<c< 1, the transformation z(z-c) w cz-1 transforms the unit circle in the z -plane into the unit circle, taken twice, in the u;-plane, and the inside of the first circle into the inside, taken twice, of the second. 26. Prove that, if a>0, maps the upper-half of the z-plane on the positive quadrant of the ttf-plane with a slit along the line v = TT, u^h 9 where w = h+in when z = I/a. (See p. 80.) 27. Show that by the transformation dz __ 1 dw ~~" the upper half of the w-plane can be mapped on the interior of a square, the length of a side of which is r J n CHAPTER IV THE COMPLEX INTEGRAL CALCULUS 30. Complex Integration The development of the theory of functions of a complex variable follows quite a different line from that of functions of a real variable. In the latter theory, having discussed functions which possess a derivative, we proceed to consider the more special class of functions which possess derivatives of the second order ; then, from among those functions which possess derivatives of all orders, we select those which can be expanded in a power series by Taylor's theorem. In complex variable theory, on the other hand, we begin by dealing with regular functions, and, by virtue of the definition of regularity, the class of functions is so restricted that a function which is regular in a region possesses derivatives of all orders at every point of the region and the function can be expanded in a power series about any interior point of the region. By following Cauchy's development of complex variable theory, everything depends upon the complex integral calculus, and, in order to prove that a regular function possesses a second derivative, we must first of all express f(z) as a contour integral round any closed contour surrounding the point z. In order to develop the subject further we must now consider the definition of the integral of a function of a complex variable along a plane curve. The equations x = <f>(t), y = iff(t) 9 where a<J<j8, define the arc of a plane curve. If we subdivide the interval 85 86 FUNCTIONS OF A COMPLEX VARIABLE (a, jS) by the points a = * , t lt J 2 , ..., t r , ..., t n = j8, then the points on the curve corresponding to these values of t may be denoted by P , P l9 P 2 , ..., P n . The length of the polygonal line PoP v ..P n , measured by r~ yr-i) 2 }*> depends on the particular r-l mode of subdivision of (a, jS). We call this summation the length of an inscribed polygon. If the arc be such that the lengths of all the inscribed polygons have a finite upper bound A, the curve is said to be rectifiable and A is the length of the curve. It can be shown that the necessary and sufficient condition that the arc should be rectifiable is that the functions <(), tfj(t) should be of bounded variation in (a, j3). If <f>'(t) and 0'() are continuous, it can be proved that the curve defined by x = <f>(t), y = *fj(t), a<tf<j8, is rectifiable and that its length s is given by * = f J If we consider an arc of a Jordan curve whose equation is z = (j)(t)-{-iifj(t), where a^^jS, we define a regular arc of a Jordan curve to be one for which <f>'(t), */*'(t) are continuous in a^^jS. From the above theorems we see that the length of this regular Jordan arc is By a contour we mean a continuous Jordan curve consisting of a finite number of regular arcs. Clearly a contour is rectifiable. We now define the integral of a function of a complex variable z along a regular arc L defined by x = <f>(t), y = <!>(*), a<*< * For proofs and further details, see P.A., p. 205 acq., or G.I., p. 113. THE COMPLEX INTEGRAL CALCULUS 87 Let f(z) be any complex function of z, continuous along L y a regular arc with end-points A and J3, and write f(z) = u(x, y)+iv(x,y). Let z , z l9 ..., z n be points on L, z being A and z n being JB. Consider the sum (Zr-*r-l)}, . . . (1) where f is any point in the arc z r _ l9 z r . If f = r 4-^ f we write u f = u( r , 7j r ), v f = v(^ r , 7j r ), and (1) may be written Now, by the mean-value theorem, X r X r _i =^(^)-^(^r-l) where ^ r _i<r r <^ f , ^ f -i<r f '<^ f . Hence the sum may be written ( Ur +iv r ){<j>'(r T ) +i^'(r r ')}(t r -t r _ l )]. . (2) r-1 Since all the functions concerned are continuous, and therefore uniformly continuous, we can, given e, find 8(e) so that for every r, provided that each | t r t r _ l |<8. Also f-1 It follows that, as and 8 tend to zero, r-1 tends to the same limit as {(* y,) r-1 88 FUNCTIONS OF A COMPLEX VARIABLE that is to the limit J Similarly the other terms of (2) tend to limits, and we find that the whole sum tends to the limit (3) This limit (3) is taken as the definition of the complex integral of f(z) along the regular arc L, and it is written f(z)dz. L The integral of /(z) along a contour (7, consisting of a finite number of regular arcs L r , is given by f f(z)dz=Z f J C r J f(z)dz. C r J L r 31. An Upper Bound for a Contour Integral I. // f(z) is continuous on a contour L, of length I, on which it satisfies the inequality \ f(z) \ ^M, then f(z)dz !, It suffices to prove this theorem for a regular arc L. Since the modulus of any integral of a function of a real variable cannot exceed the integral of the modulus of that function, we have U/w* -|J <f. a = ML If C is a closed contour we make the convention that the positive sense of description of the contour is anti-clockwise. THE COMPLEX INTEGRAL CALCULUS 89 32. Cauchy's Theorem The elementary proof of Cauchy's theorem, which depends on the two-dimensional form of Green's theorem, requires the assumption of the continuity of f'(z). We first give a proof with this assumption, but, on account of the fundamental importance of Cauchy's theorem in complex variable theory, we shall also prove the theorem under less restrictive assumptions. II. The elementary proof of Cauchy's theorem. If f(z) is a regular function and if f'(z) is continuous at each point within and on a closed contour C f , then (2)dz = (1) Let D be the closed domain which consists of all points within and on C. Then by 30 (3) we can write the integral (1) as a combination of curvilinear integrals I f(z)dz= I (udx vdy)-\-i\ (vdx -\-udy). J c J c J c We transform each of these integrals by Green's theorem,* which states that, if P(x, y), Q(x, y), dQ/8x, 8P/dy are all continuous functions of x and y in D, then Since f'(z) = u x -\-iv x = v y iu v , and, by hypothesis, f'(z) is continuous in D, the conditions of Green's theorem are satisfied and so = o, by virtue of the Cauchy-Riemann equations. * See P.A., pp. 290-1, or G.I., p. 64 90 FUNCTIONS OF A COMPLEX VARIABLE It was first shown by Goursat that it is unnecessary to assume the continuity of f'(z) and that Cauchy's theorem holds if we only assume that/'(z) exists at all points within and on C. In fact the continuity of f'(z) 9 and indeed its differentiability, are consequences of Cauchy's theorem. Second proof of Cauchy's theorem. If f(z) is regular at all points within and on the closed contour C then f f(z)dz = 0. J c The integral certainly exists, for a regular function f(z) is continuous and a continuous function is integrable. We observe also that, if we construct a network of squares, by lines parallel to the axes of x and y, having the contour C as outer boundary, then C is divided into a network of meshes, either squares or parts of squares, such that f J f(z)dz, y where y denotes the boundary of a mesh described in the same sense as C. If z lies inside a square contour 8 of side a, then l/j \z-z \ \dz\ <4v/2a 2 = 4V2(Area of S). This follows at once from 31, for | z z \<a\/2 and the length of the contour S is 4a. *^ We now prove two lemmas. Lemma 1. IfC is a closed contour, I dz = 0, zdz = 0. J c J c These results both follow from the definition of the integral, for J, n dz = lim E {{z f z f _i).\} = 0, as max | z r z r _ l \-+ 0. 7 -1 THE COMPLEX INTEGRAL CALCULUS 91 Also zdz = lim 2{z r (z r z r _J} = lim {z r -i(z f z r _i)} J c = 0, Lemma 2. Goursat's Lemma. Given , then, by suitable transversals, we can divide the interior of C into a finite number of meshes, either complete squares or parts of squares, such that, within each mesh, there is a point z such that for all values of z in the mesh, where \ e y |<. Suppose the lemma is false ; then, however the interior of C is subdivided, there will be at least one mesh for which (1) is untrue. We shall show that this necessarily implies the existence of a point within or on C at which f(z) is not differentiate. Enclose C in a large square F 9 of area A, and apply the process of repeated quadrisection. When F is quadrisected there is at least one of the four quarters of F for which (1) does not hold. Let F l be the one chosen. Quadrisect F l9 choose one quarter of F 19 and so on. We thus obtain an unending sequence of squares F l9 F 2 , ..., r n , ..., each contained in the preceding, for which the lemma is untrue. These squares determine a limit-point f , and it is clear that must lie within C. Since f(z) is differentiate at , where, for sufficiently small values of [2 |, |^|<. Now all the F r , from one particular one onwards, lie within a circle, of centre , for which | z - | is so small that | f |<. This gives a contradiction, for by taking to be z , (1) is satisfied. This proves Goursat's lemma. 92 FUNCTIONS OF A COMPLEX VARIABLE Proof of the theorem. Some of the meshes y obtained by the subdivision of the interior of G will be squares, others will be irregular, since we are not concerned with the exterior of G. Integrate (1) round the boundary of each mesh. By virtue of lemma 1, we get f(z)dz = y | z Z Q | dz\ J Y J y and so, by addition, f(z)dz = H\ Y \z-z Q \dz, J C J Y where If y is not a complete square, divide it into two parts, y x consisting of straight pieces, y a consisting of parts of (7. Since I >!<, . A being the area of the large square F surrounding G. Also, the sum of the lengths of the portions y a cannot exceed the length I of the rectifiable curve (7, and so f eylz-zJ I J Y* where K is the length of the diagonal of JT, since | z Z Q We deduce that U. f(z)dz wtere B is a constant, and, since e is arbitrary, the theorem is proved. THE COMPLEX INTEGRAL CALCULUS 93 33. Cauchy's Integral, and the Derivatives of a Regular Function By means of Cauchy's integral we can express the value of a regular function /(z) at any point within a closed contour C as a contour integral round C. III. If f(z) is regular within and on a closed contour G 9 and if be a point within (7, then Describe about z = a small circle y of radius S lying entirely within (7. In the region between C and y the FIG. 12. function (f>(z) = f(z)/(zt > ) is regular. By making a cross-cut joining any point of y to any point of C we form a closed contour F within which <}>(z) is regular, so that, by Cauchy's theorem, <l>(z)dz = 0. In traversing the contour F in the positive (counter- clockwise) sense, the cross-cut is traversed twice, once in aarh a^naa nnrl an if. fnllnwa f.hnf. each sense, and so it follows that Jo Jy Now J_ f , (z)dz I f f(*)dz = J_ f f(?)dz + JL 94 FUNCTIONS OF A COMPLEX VARIABLE Now on y, z J = 8e*0, and so the first of the two terms on the right becomes and, by theorem I, the modulus of second term on the right of (2) cannot exceed JL max |/(z)-/() | . 2778. Since /(z) is continuous at z = J this expression tends to zero as S-> : this proves the theorem. The next theorem shows how to find the value of /'() as a contour integral. IV. // /(z) is regular in a domain D, its derivative is given by o \ z ) where C is any simple closed contour in D surrounding the point z = J. We have, by III, 1 f(z)dz If we now prove that | / |-> as | ^ [-> 0, the required result is established. Since /(z) is regular in and on it is bounded, so that |/(z) [^M on (7. Let d be the THE COMPLEX INTEGRAL CALCULUS 95 lower bound of the distance of from G : suppose h chosen so small that | h \<\d, then ** 2n d*.$d 9 where I is the length of (7. It is now clear that the term on the right tends to zero as | h |-> 0. V. // f(z) is regular in a domain D, then f(z) has, at every point of D, derivatives of all orders, their values being given by If we assume the theorem proved for = m and consider the expression h we can readily prove that it is equal to (m+1)! f f(z)dz , i)i r " Jo 2ni Jc(*-) m+a ' ' and the proof that | / | tends to zero as | h \~> follows the same lines as in IV. The details are left to the reader. 34. Taylor's Theorem VI. If f(z) is regular in \ za |<p, and if is a point such that | a | = r(<p) then n-0 where a n =/ (n >(a)/n !. Let C be a circle of radius />', centre z = a, where r<p'<p, and consider the identity ^- a (t-g) 1 ^ (g-a) n - a (2-a) 2 '" (z-a) 96 FUNCTIONS OF A COMPLEX VARIABLE Multiply each term by /(z)/27rt and integrate round we clearly obtain \n i;i where f(z)dz This is Taylor's theorem with remainder. Since | /(z) | < Jf on (7 we readily see that r" where J? is a constant independent of n. Since r</>' we see that | R n |-> as n->oo. GO It therefore follows that the series 27a n (f a) n is con- o vergent and has/(^) as its sum-function. If f(z) is regular in the whole z-plane, the expansion is valid for all . Corollary. If [/() | has a maximum M(r) on | a | = r</> <Aen, t/a n =/ (n) (a)/n !, w;e Aave <Ae inequality For, if C be the circle | za \ = r we have 1 M(r) n M(r) 35. The Theorems of Liouville and Laurent VII. Liouville's Theorem. // /(z) is regular in the whole z-plane and if |/(z) | <K for all values of z, then /(z) must be a constant. Let z l9 z 2 be any two points and a circle of centre z l THE COMPLEX INTEGRAL CALCULUS 97 and the radius p>2 |2 1 z 2 1, so that, when z is on (7, . By III, so that - * i r fa-sj/tt* < i r - 0^1 J c (z_ Zl)(2 _ 22 ) < 27 r J Keep 2, and z 2 fixed and make p->oo, then it follows that /(Zj) = /(z 2 ) ; in other words, /(z) is a constant. VIII. Laurent's Theorem . Let C l and C 2 be two circles of centre a with radii p l and p 2 (p 2 <Pi) 5 th en > tff( z ) be regular on the circles and within the annulus between (/j and <7 a , J being any point of the annuliis. The coefficients a n and b n are given by By making a cross-cut joining any point of O x to any point of C t , we readily see that (z)dz Consider the two identities JL s J_j._iz^^. - (C-^)"- 1 , (C-) n i - z-a' t "(2-a) a " f "*"" r " ( 2 -o) "" (z-a)" *- z - 2 -""- 1 z - n i 98 FUNCTIONS OF A COMPLEX VARIABLE O^n it Mows that where (z-a)(z--)' 27* Now P n is precisely the same remainder as in Taylor's theorem and we can prove, in the same way as in VI, that | P n |-> as Also we have f f^Y&^ Jc.l^-a/ -{ where r = | ^ a | and |/(z) it follows that | Q n |-> as U-+CQ If, therefore, we write ' on (7 2 . Since /) a <r, where ^(f) = a n (-a) and /,() = MC-a)-, we o o see that/(^) converges forp 2 ^| ^ a |^/)j. It also follows that/ 1 (^) is regular and converges for | fl |^PI and that / a () is regular and converges for 36. Zeros and Singularities If f(z) is regular within a given domain Z), we have seen that it can be expanded in a Taylor series about any point z = a of D and THE COMPLEX INTEGRAL CALCULUS 99 If a Q = a l = ... = a m-1 = 0, a m 4= 0, the first term in the Taylor expansion is a m (z a) m . In this case f(z) is said to have a zero of order m at z = a. A singularity of a function f(z) is a point at which the function ceases to be regular. If /(z) is regular within a domain D, except at the point z = a, which is an isolated singularity of /(z), then we can draw two concentric circles of centre a, both lying within D, The radius of the smaller circle p a may be as small as we please, and the radius p l of the larger circle of any length, subject to the restriction that the circle lies wholly within D. In the annulus between these two circles, /(z) has a Laurent expansion of the form /(z) = f a n (z-a)+f b n (z-a)~\ o i The second term on the right is called the principal part of /(z) at z = a. It may happen that b m * while 6 m+1 = 6 m + 2 ... = 0. In this case the principal part consists of the finite number of terms and the singularity at z = a is called a pole of order m of /(z) and the coefficient b l9 which may in certain cases be zero, is called the residue of /(z) at the pole z = a. If the pole be of order one, b l = lim{(z a)/(z)}. If the principal part is an infinite series, the singularity is an isolated essential singularity. (1) If z = a is a zero of order m of /(z), we now prove that this zero is isolated : in other words, there exists a neighbourhood of the point z = a which contains no other zero off(z). Clearly we can write /(z) = (z a) m ^(z), where ^(z) is regular in | a \<p and <j>(a) * 0, since </>(a) = a m . 100 FUNCTIONS OF A COMPLEX VARIABLE Write ^(a) =s 2c, then, since <f>(z) is continuous, there exists a region | z a |<8 in which |^(z) ^(a) | < | c \. Hence where | za |<8, and so ^(z) does not vanish in | z a |<8. (2) If z = a is a pole of order m of /(z) it follows, from the definition of a pole by means of Laurent's theorem, that poles are isolated, for, the small circle, of centre z = a and radius p a , encloses the only singularity of /(z) within the domain D which contains the annulus between the two circles of radii p l and p a . (3) ///(z) has a pole at z = a, ften |/(z) [->oo a$ z-* a tn ant/ manner. For, if the pole be of order m, f(z) = (-a)--{6 m +6 m . 1 (z~o)+...+6 1 (-ar-Hra n (2~ar^} o and, since 6 m * 0, we may write /(z) = (z a)- m ^r(z), where ^(z) is regular in | za [</>, and 0(a) = 6 m ( 4 s 0). Hence, by (1), we can find a neighbourhood | za |<8 of the pole in which | ^r(z) |>J | 6 m |, from which it follows that l/(*)l>ilMI*-*|- m . Hence |/(z) |-> co as z-> a in any manner. (4) Limit points of zeros and poles. Let a x , a 2 , ..., a n , ... be a sequence of zeros of a function /(z) which is regular in a domain D. Suppose that these zeros have a limit point a which is an interior point of D. Since /(z) is a continuous function, having zeros as near as we please to a, /(a) must be zero. Now z = a cannot be a zero of /(z), since we have proved in (1) that zeros are isolated. Hence /(z) must be identically zero. If /(z) is not identically zero in Z), then z = a must be a singularity of /(z). The singularity is isolated, but it is not a pole, since |/(z) | does not tend to infinity as z-+ a in any manner. Hence a limit point of zeros must be an isolated essential singularity of /(z). THE COMPLEX INTEGRAL CALCULUS 101 If f(z) be regular, except at a set of points which are singularities c l9 c a , ..., c n , ..., infinite in number, and having a limit point y in D, then y must be a singularity of/(z), since /(z) is unbounded in the neighbourhood of y. Since y is not isolated it cannot be a pole. We call such a singularity a non-isolated essential singularity. Examples, sin 1/z has an isolated essential singularity at z = 0. It is the limit point of the zeros, z = l/n?r, (n = J^l, 2, ...). tan 1/2 has poles at the points z = 2/nrr, (n = 1, 3, ...)> and so the limit point of the poles, z = 0, is a non- isolated essential singularity. Note on the region of convergence of a Taylor series. Iff(z) be a function which is regular, except at a number of isolated singularities at finite points of the z-plane, then we can expand f(z) in a Taylor series Sa n (za) n o about any assigned point z = a, and the radius of con- vergence p of this power series will be the distance from z = a to the nearest singularity of/(z), since /(z) is clearly regular in | z a |<p, and cannot be regular in any circle of centre a whose radius exceeds p. We see that the radius of convergence of a power series depends upon the extent of the region within which the sum-function is regular, and it may be controlled by the existence of singularities which do not necessarily lie on the real axis. If we consider the real function 1/(1 a), the binomial expansion leads to the series being convergent if | x \<1 This seems quite natural, since the sum-function has a singularity at x = 1. However, on considering the function l/(l+x a ), we have ..... (2) 102 FUNCTIONS OF A COMPLEX VARIABLE and the series is again convergent only if | x |<1 ; but if we regard l/(l+# 2 ) as a function of the real variable x there is nothing in the nature of the function to suggest the restriction of its range of convergence to |#|<1. If, however, we consider l/(l+z 2 ), where z is complex, the restriction on the region of convergence is at once evident, since l/(l+z 2 ) has singularities at z = , and the radius of convergence of the series (2), if a; is complex, is the distance of the origin from the nearest singularity and this is plainly unity. 37. The Point at Infinity In complex variable theory we have seen that it is convenient to regard infinity as a single point. The behaviour of /(z) "at infinity " is considered by making the substitution z = l/ and examining /(!/) at = 0. We say that /(z) is regular, or has a simple pole, or has an essential singularity at infinity according as /(!/) has the corresponding property at = 0. We know that if /(!/) has a pole of order m at f = 0, near = we have n-O 2 b and so, near z = oo, /(z) = 2 a n z~+b l z+b 2 z*+...+b m z<. n-O Thus, when /(z) has a pole of order m at infinity, the principal part of /(z) at infinity is the finite series in ascending powers of z. Since the function sinz has an isolated essential singularity at infinity, the principal part at infinity being an infinite series. THE COMPLEX INTEGRAL CALCULUS 103 38. Rational Functions Theorem. If a single-valued function f(z] lias no essential singularities either in the finite part of the plane or at infinity, thenf(z) is a rational function. Since the point at infinity is not an essential singularity of f(z), we can surround it by a region in which f(z) either is regular or has the point at infinity as its only singularity. That is, we can draw a circle (7, with centre the origin, such that the point at infinity is the only singularity outside G. There can only be a finite number of singularities within C, since poles are isolated singularities. Suppose that the poles inside G are at a v a 2 , ..., a n . The principal part at a f may be written '28 z-a, ^ (z-a,) 2 ^ " ^ (z-a,)-' a, being supposed to be a pole of order m. The principal part at infinity is of the form Now consider the function The function ^(2) is plainly regular everywhere in the plane, even at infinity : hence <f>(z) is bounded for all z, and so, by Liouville's theorem, <f>(z) is a constant. Hence f(z) - 0+2 and so /(z) is a rational function of 2. 39. Analytic Continuation Suppose that / x (z) and / 2 (z) are functions regular in domains D l and J5, respectively and that D l and D 2 have 104 FUNCTIONS OF A COMPLEX VARIABLE a common part, throughout which f^z) = / a (z), then we regard the aggregate of values of f^(z) and / 2 (z) at points interior to D l or D 2 as a single regular function (f>(z). Thus <f>(z) is regular in A = D^-\-D^ and <f>(z) = / 1 (z) in JDj and <f>(z) = / 2 (z) in Z) 2 . The function / 2 (z) may be regarded as extending the domain in which f^z) is defined and it is called an analytic continuation of /^z). The standard method of continuation is the method of power series which we now briefly describe. Let P be the point z , in the neighbourhood of which f(z) is regular, then, by Taylor's theorem, we can expand f(z) in a series of ascending powers of z z , the coefficients in which involve the successive derivatives of f(z) at z . If S be the singularity of f(z) which is nearest to P, then the Taylor expansion is valid within a circle of centre P and radius PS. Now choose any point P l within the circle of convergence not on the line PS. We can find the values of f(z) and all its derivatives at P l9 from the series, by repeated term-by-term differentiation, and so we can form the Taylor series for f(z) with P l as origin, and this series will define a function regular in some circle of centre P v . This circle will extend as far as the singularity, of the function defined by the new series, which is nearest to P l and this may or may not be S. In either case the new circle of convergence may lie partly outside the old circle and, for points in the region included in the new circle but not in the old, the new series may be used to define the values of f(z) although the old series failed to do so. Similarly, we may take any other point P 2 in the region for which the values of the function are now known and form the Taylor series with P 2 as origin which will, in general, still further extend the region of definition of the function ; and so on. By means of this process of continuation, starting from a representation of a function by any one power series, we can find any number of other power series, which THE COMPLEX INTEGRAL CALCULUS 105 between them define the value of the function at all points of a domain, any point of which can be reached from P without passing through a singularity of the function. It can be proved that continuation by two different paths PQR, PQ'R gives the same final power series provided that the function has no singularity inside the closed curve PQRQ'P. We may now, following Weierstrass, define an analytic function of z as one power series together with all the other power series which can be derived from it by analytic continuation. Two different analytic expressions then define the same function if they represent power series derivable from each other by continuation. The complete analytic function defined in this way need not be a one- valued function. Each of the continuations is called an element of the analytic function. If f(z) is not an integral function there will be certain exceptional points which do not lie in any of the domains into which /(z) has been continued. These points are the singularities of the analytic function. Clearly the singular points of a one- valued function are also singularities in this wider sense. There must be at least one singularity of the analytic function on the circle of convergence (7 of the power 00 series a n (z z ) w .* For, if not, we could construct, by o continuation, a function equal to /(z) within <7 but regular in a larger concentric circle jT . The expansion of this function in a Taylor series in powers of z z would then converge everywhere within F Q . This is impossible, since the series would be the original series whose circle of convergence is (7 . If z l is any point within <7 , let C l be the circle of convergence of the power series * For a proof of this, and further details, see Titchmarsh, Theory of Function* (Oxford, 1932), p. 145. 106 FUNCTIONS OF A COMPLEX VARIABLE If Q l is the circle of centre z l which touches (7 internally, the new power series is certainly convergent within Q l and has the sum /(z) there. There are now three possibilities. Since the radius of C l cannot be less than that of Q v we have either (i) C l has a larger radius than Q l9 or (ii) (7 is a natural boundary * of /(z), or (iii) C l may touch (7 internally, though (7 is not a natural boundary of/(*). In case (i) C l lies partly outside (7 and the new power series provides an analytic continuation of /(z) : we can then take a point z a within C l and outside C and repeat the process. In case (ii) we cannot continue /(z) outside (70 and the circle C l touches (7 internally no matter what point z l within (7 is chosen. In case (iii) the point of contact of (70 and G l is a singularity of the analytic function obtained by continuation of the original power series. For there is necessarily one singularity on C l and this cannot be within (7 . We may illustrate some of the above remarks by the following examples. 1. The series represents the function f(z) = l/(az) only for points within the circle \z\ =* | a |. If 6/a is not real, the series 1 *-b (s-6) a-6 + (a-6) 1 "*" (a~6) "*" "' f for different values of 6, represents f(z) at points outside the circle | z \ = | a |. 2. That there are functions to which the process of continuation cannot be applied may be seen by considering the function g(z) = l+2 2 +z* + ...+* |n + .... * See Example 2 below. THE COMPLEX INTEGRAL CALCULUS 107 It is readily shown that any root of any of the equations 2 1 1 2 4 1 * 1 ! I Z - 1, Z - 1, 1, 9 ... 9 is a singularity of g(z) 9 and hence that on any arc, however small, of the circle | z \ = 1 there is an unlimited number of them. The circle | z \ = 1 is in this case a natural boundary of g(z). This illustrates case (ii) above, 40. Poles and Zeros of Meromorphic Functions A function /(z), whose only singularities in the finite part of the plane are poles, is called a meromorphic function. We now prove a very useful theorem. // /(z) is meromorphic inside a closed contour C, and is not zero at any point on the contour, then *- - <" where N is the number of zeros and P the number of poles inside 0. (A pole or zero of order m must be counted m times.) Suppose that z = a is a zero of order w, then, in the neighbourhood of this point /(z) = (z-a)^(z), where <f>(z) is regular and not zero. Hence "*" __ /(z) z-a "" ftz) ' Since the last term is regular at z = a, we see that /'(z)//(z) has a simple pole at z = a with residue m. Similarly, if z = 6 is a pole of order k, we see that /'(z)//(z) has a simple pole at z = 6 with residue ft. It follows, by 33, III, that the left-hand side of (1) is equal to ZmSk = NP. If /(z) is regular in (7, then P = 0, and the integral on the left of (1) is equal to N. Since 108 FUNCTIONS OF A COMPLEX VARIABLE we may write the result in another form, W) dz==Al where A c denotes the variation of log/(z) round the contour C. The value of the logarithm with which we start is immaterial ; and, since and log |/(z)| w one- valued, the formula may be written N=-A *Tgf(z). This result is known as the principle of the argument. 41. Rouch6 's Theorem If f(z) and g(z) are regular within and on a closed contour C and | g(z) \ < \ f(z) \ on C, then f(z) and f(z)+g(z) have the same number of zeros inside C. We observe that neither f(z) nor f(z) +g(z) has a zero on (7, and so, if N is the number of zeros of f(z) and N' the number of zeros of f(z)+g(z), 2<jrN = A c arg/, 27rN' = AC arg (f+g) = A Q arg /+ A arg ( J + y) The theorem is proved if we show that +j\ = 0. Since | g \ < |/|, the point w = l+g/f is always an interior point of the circle of centre w = 1 and radius unity : thus, if w = pety, <f> always lies between \n and \n and so arg (1 +g/f) = ^ returns to its original value when z describes C. Since <f> cannot increase or decrease by a multiple of 2rr, the theorem follows. THE COMPLEX INTEGRAL CALCULUS 109 The preceding theorems are useful for locating the roots of equations. The method is illustrated by the following example. Example. Prove that one root of the equation z 4 +z 8 + l =0 lies in the first quadrant. The equation z 4 +z 3 + l = plainly has no real roots. For, if we put z = x, x*+x* + l = has no real positive roots. If we put z = a; and write $(x) = x* # 3 + 1=0, we see that , if x>\ ; and <f>(x) =x*+(l~x)(x*+x + l)>0 9 if Hence the given equation has no real negative roots. The given equation has no purely imaginary roots either, for, on putting z = iy, we get y 1 iy* + l = and it is plain that the real and imaginary parts never vanish together. Consider A arg (z 4 +z 8 + l) round part of the first quadrant bounded by | z \ = R where R is large. On the arc of the circle, z = ReiO, and we have A arg ( 2 4+z 8 + l) = j arg (R* e *iO)+ A arg {1 +O(R~ 1 )}, = 2<jr On the axis of y we have arg(z 4 +z s + l) = arc tan ~~ y The numerator of y*/(y l + l) only vanishes when y = and the denominator does not vanish for any real y. Hence as y ranges from oo to along the imaginary axis, the initial and final values of arg (z 4 +z a + l) are zero. Hence the total change in arg(z 4 +z 8 + l), where R is large, is 2w. It follows that one root of the given equation lies in the first quadrant. 42. The Maximum-Modulus Principle We now establish an important theorem which may be stated as follows. // /(z) is regular within and on a dosed contour C 9 then |/(z)| attains its maximum value on the boundary of G and not at any interior point. 110 FUNCTIONS OF A COMPLEX VARIABLE Lemma. If<f>(x) is continuous, (f>(x)^,K and . (1) then (f>(x) = K. Suppose that ^(x 1 )</c, then there is an interval (x l 8, Zx+S) in which </>(X)^.K and which contradicts (1). Theorem. If |/(z)|<Jf on C, then \f(z)\<M at all interior points of the domain D enclosed by C, unless f(z) is a constant j in which case \f(z)\ = M everywhere. Suppose that at an interior point z of D, \f(z)\ has a value at least equal to its value elsewhere. Let F be a circle of centre Z Q lying entirely within D. Then by 33, in, /w _ ' r /w* . . . (2) 2<TTl J r Z-Z Q Write z z = re^, f(z)/f(z Q ) = petf, so that p and ^ are functions of 0, then (2) may be written 1 f2 = ^ J o MM. . . . (3) Hence 1 < ^- I odd. By hypothesis p^l, and so, by the lemma, p = 1 for all values of 0. On taking the real part of (3) we get THE COMPLEX INTEGRAL CALCULUS II! and so, by the lemma, cos^i = 1. Hence f(z) =/(z ) a JT. Since f(z) is a constant at any point a on J*, it follows by Taylor's theorem that it is constant in a neighbourhood of a, and hence, by analytic continuation, f(z) is constant everywhere within and on C. There is a corresponding theorem for harmonic functions. A function which is harmonic in a region cannot have a maximum at an interior point of the region. EXAMPLES IV 1. The function /(z) is regular in | z a \<R ; prove that, if 0<r<R, I /-27T .- f'(a) - _ P(e)e-* dO, 7TT J Q where P(0) is the real part off(a+reiO). 2. <f>(z) and $(z) are two regular functions ; z = a is a once repeated root of $(z) = and <f>(a) ^ 0. Prove that the residue of <t>(z)l\ff(z) at z = a is {6 ^'(a)0"(a) -2 #a)0'"(a)}/3 {f' (a)}. 3. The function f(z) is regular when | z \<R'. Prove that, if | a \<K<R', where O is the circle | z \ = B. Deduce Poisson's formula that, if 0<r<K, 1 T 27r # a r* T, / *-2Kr C o* ( e-t )+ r* 4. By using the integral representation of / (n) (a), ( 33, V), prove that ajfl \ t _ 1 f x " e *' fTl) = 2^ J c nTz*+* *' where C is any closed contour surrounding the origin. Hence prove that 112 FUNCTIONS OF A COMPLEX VARIABLE 6. Obtain the expansion KM} /V; - ... 2 J \ 2 / + 2>.3t / \2/^2.6! / \2/^ J and determine its range of validity. 6. If f(z) = S z a /(4+n 2 z a ), show that f(z) is finite and w*i continuous for all real values of z but/(z) cannot be expanded in a Maclaurin series. Show that /(z) possesses Laurent expansions valid in a succession of ring spaces. 7. Prove that cosh (z+-j = a + a n (z n H ) , where \ zl i \ z n l 1 r 27r o n == cos n6 cosh (2 cos 8)dd. 2n J 8. Find the Taylor and Laurent series which represent the function (z*-l)/{(z+2)(z+3)} in (i) | z |<2, (ii) 2<| 2 |<3, 9. Find the nature and location of the singularities of the function l/{z(e f 1)}. Show that, if 0< | z |<27r, the function can be expanded in the form and find the values of a and a a . 10. The only singularities of a single-valued function f(z) are poles of orders 1 and 2 at z = 1 and z = 2, with residues at these poles 1 and 2 respectively. If /(O) = 7/4, /(I) = 6/2, determine the function and expand it in a Laurent series valid in 1< | z |< 2. 11. Classify the points z = 0, z = 1 and the point at infinity, in relation to the function /(,).!=? sin jij, and find the residues of /(z) at z = and at z = 1. 12. Show that, if 6 is real, the series THE COMPLEX INTEGRAL CALCULUS 113 is an analytic continuation of the function defined by the series 13. The power series z + z 2 +i^ 8 + and have no common region of convergence : prove that they are nevertheless analytic continuations of the same function. 14. If a>e, use Rouch6's theorem to prove that e* = az* has n roots inside the circle | z \ = 1. 15. The Fundamental Theorem of Algebra. By taking f(z) = a z m , g(z) = c^z- 1 +a^z m ~ z + ... +a m , use RoucWs theorem to prove that the polynomial F(z) = a z"+a 1 z'- 1 + ... +a m has exactly m zeros within the circle \z\ = R for sufficiently large R. Deduce from Liouville's theorem that F(z) has at least one zero. 16. Prove that 3 8 +3z 8 +7z+6 has exactly two zeros in the first quadrant. 17. If |/(z) \>m on | z \ = a, /(z) is regular for | z \^a and |/(0) | <ra, prove that /(z) has at least one zero in | z | <a. (See 42.) Deduce that every algebraic equation has a root. (This is another proof of the Fundamental Theorem of Algebra.) 18. If a domain D of the z -plane is bounded by a simple closed contour C and w =/(z) is regular in Daiid on C7, prove that, if /(z) takes no value more than once on C, then f(z) takes no value more than once in D. (Use the theorem of 40.) Prove that the above result holds for the function w = z a +2z+3, if D is the domain | z \ < 1 and C is the unit- circle. CHAPTER V THE CALCULUS OF RESIDUES 43. The Residue Theorem We now turn our attention to the residue theorem, and to one of the first applications which Cauchy made of this theorem the evaluation of definite integrals. It should be observed that a definite integral which can be evaluated by Cauchy's method of residues can also be evaluated by other means, though usually not so easily.* We have already defined the residue of a function f(z) at the pole z = a to be the coefficient of (z a)~ l in the Laurent expansion of /(z), which, if z = a is a pole of order m, takes the form Za n (z-a)+3b n (z-a)-*. o i We have also remarked that, when z = a is a pole of order one, the residue b l can be calculated as lim{(z a)/(z)}. ->a The residue can also be defined as follows. If the point 2 = a is the only singularity of /(z) inside a closed contour 1 f C y and if . I f(z)dz has a value, that value is the residue 1m J Q of /(z) at z = a. The residue of /(z) at infinity may also be defined. If /(z) has an isolated singularity at infinity, or is regular /oo e~ m * dx, sometimes stated to be an integral which cannot be evaluated by Cauchy's method, see Courant, Differential and Integral Calculus, II, p. 661. In this case Cauchy's method is the more difficult. 114 THE CALCULUS OF RESIDUES 115 there, and if G is a large circle which encloses all the finite singularities of/(z), then the residue at z = oo is defined to be taken round O in the negative sense (negative with respect to the origin), provided that this integral has a definite value. If we apply the transformation z = l/ to the integral, it becomes taken positively round a small circle, centre the origin. It follows that if has a definite value, that value is the residue of /(z) at infinity. Note that a function may be regular at z = oo but yet have a residue there. The function /(z) = A/z has a residue A at z = and a residue A at z = oo, al though /(z) is regular at z = oo. Theorem 1. Cauchy's Residue Theorem. Let f(z) be continuous within and on a closed contour and regular, save for a finite number of poles, within G. Then I f(z)dz = 2mSy%> J c where Z72 is the sum of the residues off(z) at its poles within C. Let a lf a 2 , ..., a n be the n poles within C. Draw a set of circles y r of radius 8 and centre a r , which do not intersect and which all lie inside O. Then /(z) is certainly regular in the region between C and these small circles y f . 116 FUNCTIONS OF A COMPLEX VARIABLE We can therefore deform C until it consists of the small circles y r and a polygon P which joins together the small circles as illustrated in fig. 13. FIG. 13. Then f f( z )dz = f f(z)dz+Z ( f(z)dz = 2 f f(z)dz, J C J P fl J 7r f = l J Yr for the integral round the polygon P vanishes because f(z) is regular within and on P. If a f is a pole of order m, then where <f>(z) is regular within and on y f . Hence f m C b f(z)d*=Z 7-~T^Z. J y r 8-1 J y r \ z a r) On writing z = a r -\-&eW, varies from to 77 as the point z makes a circuit of the circle y r , and so f f( z )dz = Zb 8 &-> \ * e(i-'Wid0 = 2irib v J Yr i J Hence j f(z)dz = Z \ f(z)dz = 2iri J C f-l J Yr which proves the theorem. THE CALCULUS OF RESIDUES 117 Theorem 2. // lim{(z a)f(z)} = 6, and if C is the i-^a arc, #!< arg (2 a) <0 2 , o/ fe circle \ za \ = r, lim f /(z)dz r-^Oj Given c, we can find an 77 (e) such that, if | za |<ij, | S |<, where (z-a)/(z) = fr+8. Hence f // 1 J f 6 + 8 J f *' /( 2 ) <fe = dz = Jo Jcz o Jfli i r IJo and so, on taking the limit as r-> 0, the theorem follows. If 2 = a is a simple pole of /(z), 6 is the residue of f(z) at z = a, so that if the contour is a small circle surrounding the pole, a #1 == 2ir and we get J, /(Z) dz = 27N&. 44. Integration round the Unit Circle We consider first the evaluation by contour integration of integrals of the type f , sin0)d0, where <(cos0, sin#) is a rational function of sin0 and cos 0. If we write z = e*0, then f27T f and so I ^(cos 0, sin 0)d0 = 4t(z)dz, Jo Jo 118 FUNCTIONS OF A COMPLEX VARIABLE where 0(z) is a rational function of z and is the unit circle I z I = 1. Hence /.< where Z7c denotes the sum of the residues of ^r(z) at its poles inside C. Example. Prove that, if a > 6 > 0, J = \ Now on making the above change of variable, if C is the unit circle | z \ = 1, j = JL J _XI l^ZI = J_ I _JL: i^I_ = 1 f ^( Z )^, where are the roots of the quadratic z*+2az/b + l = 0. Since the product of the roots a, j8 is unity, we have | a || ]8 | = 1 where | ft \>\ a |, and so z = a is the only simple pole inside C. The origin is a pole of order two. We calculate the residues at (i) z = a, and (ii) z = 0. (a--)' (i) Residue = lim(z-a)F(z) = lira ~ ^-' = - - ^ z-+a z^a z ( z -P> a ~P (z I) 1 (ii) Residue is the coefficient of 1/z in - - - , Z \Z "T~iC**y(/ "|~ L) where z is small. Now z\z* + 2az/b + 1 ) and coefficient of l/z is plainly 2a/6. THE CALCULUS OF RESIDUES 119 which proves the result. 45. Evaluation of a Type of Infinite Integral Let Q(z) be a function of z satisfying the conditions : (i) Q(z) is meromorphic in the upper half-plane ; (ii) Q(z) has no poles on the real axis ; (iii) zQ(z)-+ uniformly, as | z |->oo, for 0<arg ( f [0 iv) I Q(x)dx and Q(x)dx both converge ; then Jo J oo where EJ& denotes the sum of the residues of Q(z) at its poles in the upper half-plane. Choose as contour a semicircle, centre the origin and radius R, in the upper half-plane. Let the semicircle be denoted by I 7 , and choose R large enough for the semicircle to include all the poles of Q(z). Then, by the residue theorem, f Q(x)dx+ f Q(z)dz = J -R J r From (iii), if R be large enough, | zQ(z) |< for all points on JT, and so I f Q(z)dz = I { V Q(Ref8)R&OidO j <c f" d0 = TIC. Hence, as jR->oo, the integral round JT tends to zero. If (iv) is satisfied, it follows that /)//M\/7/M _ OUM|* ^*f&^ \\X](LX = ^TTl^i//^^. oo If Q(z) be a rational function of z, it will be the ratio of two polynomials N(z)/D(z), and condition (iv) is satisfied 120 FUNCTIONS OF A COMPLEX VARIABLE if the degree of D(z) exceed that of N(z) by at least two, for, when x is large, Q(x) behaves like x~* 9 where p^2 and f dx A f ~ M dx i. *u * * I and I both exist.* J A x> J -*x* Note that condition (iv) is required as well as (iii), for the condition xQ(x)->0 is not in itself sufficient to secure the /GO convergence of I Q(x)dx. This can be seen by taking Q(x) = (* log *)-*. Example. Prove that, if a>0, dx If 2 4 +a* = 0, we have z 4 == a 4 e 7ri , and the simple poles of the integrand are at ae 7 ^/ 4 , ae 377 */ 4 , ae 57r ^/ 4 , ae 77ri / 4 . Of these, only the first two are in the upper half-plane. The conditions of the theorem are plainly satisfied, and so dx = 2iri E {Residues at z = ae 77 */ 4 , ae 37r */ 4 }. Let k denote any one of these, then A 4 = a 4 and the residue at the simple pole z = k is lim{(2 fc)(z 4 & 4 )- 1 }. This may *->* be evaluated by Cauchy's formula, as applied to the evaluation of limits of expressions of the indeterminate form f 0/0, and so z-k Hence * For the convergence of infinite integrals, see P. A., p. 193, or G.I., p. 77. t P.A., p. 106. THE CALCULUS OF RESIDUES 121 00 dx f ax Hence I . = j x*+a 4 The theorem can be extended to the case in which D(z) = has non-repeated real roots, so that Q(z) has simple poles on the real axis. We now indent the contour by making small semicircles in the upper half-plane to cut out the simple poles on the real axis. Suppose that D(z) = has only one root z = a, where a is real. The contour is then as shown in fig. 14. The small semicircle is denoted by y, its centre is the point x = a and its (small) radius is />. If F is large enough to enclose all the poles of Q(z) in the upper half-plane, then the integral round F tends to zero as R-+OO, as before. We therefore have, if the path of integration be as indicated by the arrows in fig 14, a+p f + P~ P + f + f* J T J -R Jy Ja fa-p [R f As IZ^oo, + Q(x)dx = P Q(x)dx, J -R J a+p J -oo and it remains to consider I Q(z)dz. Now, on y f JY z = a+/>e l *0, and so f Q(z)dz= t J y J 122 FUNCTIONS OF A COMPLEX VARIABLE * Since Q(z) contains the factor (2 a)- 1 we may write Q(z) = <f)(z)I(za) and ^(z) is regular at and near z = a. Hence f Q(z)dz = f <(>(a+peiO)id6 = t f J y J 7T J since <f>(a+peiO) is regular at and near a and can be expanded by Taylor's theorem with remainder when n = 1. It follows that Q(z)dz-+ 7n^(a) as p-> 0. y Since ^(a) is plainly the residue of Q(z) = <f>(z)/(za) at z = a, we can write the final result in the form f J -0 Q(x)dx where Z7 denotes the sum of the residues of Q(z) at its simple poles on the real axis, for clearly each pole on the real axis can be treated in the same way as z = a. The principal value of the integral is involved, because equal spaces p are taken on either side of the real poles, and, by definition,* lim (~~ P + ( ft f(x)dx = P ( ft f(x)dx. p-+Q J a J a+p J a It should be noticed that if a pole be cut out by a small semicircle, the contribution to the value of the integral is half what it would be if a small circle surrounded the pole. (See 43, Theorem 2.) 46. Evaluation of Infinite Integrals by Jordan 's Lemma We now prove a very useful theorem which is usually known as Jordan's lemma. * See P.A., p. 195, or G.I., p. 81. THE CALCULUS OF RESIDUES 123 If F be a semicircle, centre the origin and radius R, and f(z) be subject to the conditions : (i) f(z) is meromorphic in the upper half -plane, (ii) /(z)-> uniformly as \ z |-*oo/or 0<arg (ill) m is positive ; then J. e mit f(z)dz-+ as J?->oo. By (ii), if R is sufficiently large, we have, for all points on/ 1 , | /(z)|<. Now | exp miz \ = | exp{wiJ?(cos0+t sin0)}| = Hence, j f(z)e*dz = I f(z)e>i*ReiOid6 < J e-Ri J 2Re * Now it can be proved, by considering the sign of its derivative, or otherwise, that sinO/d steadily decreases from 1 to 2/7T as 6 increases from to Jw. Hence, if sin0 ~W Hence I f /(zjgm&dz < 2Re ( e-2M/nd0, m m from which the lemma follows. By using this lemma, we can evaluate another type of integral. The method may be set out as a theorem as follows. 124 FUNCTIONS OF A COMPLEX VARIABLE Let Q(z) = N(z)/D(z), where N(z) and D(z) are poly* nomials, and D(z) = has no real roots, then if (i) the degree of D(z) exceeds that of N(z) by at least one, (ii) m>0, f00 Q(x)e mi *dx = I. where 27S+ denotes the sum of the residues of Q(z)e mi9 at its poles in the upper half -plane. If we write f(z) = Q(z)e m< * , we see that f(z) satisfies the conditions of Jordan's lemma and so I f(z)dz-> as jR->oo. On using the same contour as before, a large semicircle in the upper half-plane, by making B-+CQ we get Q(x)e mix dx = ^mETe*. 30 On taking real and imaginary parts of this result we see that by this method we can evaluate integrals of the type I f(x) cos mx dx , f(x) sin mx dx. J -00 J -00 By a well-known test for convergence of infinite in- fx tegrals,* if f(x) decreasesand -> as s->oo, since , c ? 8 mx dx J a sm is bounded, the integrals in question converge. Example. Prove that, if a>0, m>0, 00 cosma;^ ir f J The only pole of the integrand considered, e in the upper half-plane, is a double pole at z = ai. The * The test is known as Dirichlet's test. See Titchmarsh, Theory of Functions (Oxford, 1932), p. 21. THE CALCULUS OF RESIDUES 125 conditions of the theorem are easily seen to be satisfied and so dx =s 27rt{Residue of at z = ai}. jo (a 2 +a; 2 ) a (a 2 +z 2 ) a Put z = ai+t then, since t is small, gtnif e~ mo 6 m ** 6""*** (a a +z a ) a < 2 (2at+0 1 4a 2 * 2 and the residue, which is the coefficient of tr l , is easily seen to be te- ma (l+ma) f e m<a! dx 7re~ tno (l+ma) Hence = ^ ' . J -oo (a 2 +x 2 ) 2 2a 8 On equating real parts, and taking one half of the result for the integral from to oo, we get r cos mx dx TJ If there are simple poles of the integrand on the real axis, we get a modification of this result, similar to that obtained for the theorem of 45. Thus, if D(z) = has non-repeated real roots, we get r 00 > I 0(x )e mi *dx = the proof following the same lines as before. An example of this extension of the theorem is the proof that 00 * ! dx = JTT, if W>0. On considering the integrand e mi '/z we see that it has a simple pole at z = and none in the upper half -plane. The residue at z = is easily seen to be unity, and so we get r oo 0m< j 1 J -at X 126 FUNCTIONS OF A COMPLEX VARIABLE On equating real and imaginary parts we get cos mx _ dx = 0, * sin mx _ dx = ir i r J the " P '* is not necessary in the second integral, since sin mx/x^- m as #-> 0, whereas the integrand in the first integral becomes infinite at the origin. From the second result we get 00 sin mx , dx = JTT. x 47. Integrals Involving Many- Valued Functions f A type of integral of the form &- l Q(x)dx, where a Jo is not an integer, can also be evaluated by contour integration, but since z a ~ l is a many-valued function, it becomes necessary to use the cut plane. One method of dealing with integrals of this type is to use as contour a large circle F, centre the origin, and radius B ; but we must cut the plane along the real axis from to oo and also enclose the branch-point z = in a small circle y of radius p. The contour is illustrated in fig. 15. Fio. 15. THE CALCULUS OF RESIDUES 127 Let Q(x) be a rational function of x with no poles on the real axis. If we write f(z) = z?~ l Q(z) and suppose that z/(z)-> uniformly both as | z \ -> and as | z \ ->oo, then we get the integral round F tending to zero as jR->oo and the integral round y tending to zero as />-> ; for, on F, if -R is large enough, | zf(z) \ <e and so \f(z) \ </R 9 \z)dz Similarly on y, | z/(z)|< if p is small enough, and so f(z)dz < - 2np = 27T. P Hence on making p-+ and R-+CQ we get /> fo &~ l Q(x)dx + JO Jo where Sy\* is sum of residues of /(z) inside the contour. We observe that the values of z*" 1 at points on the upper and lower edges of the cut are not the same, for, if z = re^, we have z*"" 1 = r*- V^*" 1 * and the values of z at points on the upper edge correspond to | z | = r, = 0, and at points on the lower edge they correspond to | z \ = r , = 277. Since e^wifa"" 1 ) - ^Trtc^ \^e get f J We also observe that, when calculating the residues at the poles, z a ~~ l must be given its correct value r a at each pole. Example. Prove that f &- l dx ir .. _ -7- = -: , if 0<o<l. Jo 1 +a? sin aw 128 FUNCTIONS OF A COMPLEX VARIABLE Here we observe that, when/(z) = z?~ l (l +z)~ l , z/(z)-> aa | z | ->oo, if a< 1, and z/(z)-> as | z \ -> 0, if a>0. Hence, if 0<a<l, the integral round F tends to zero as .R->oo and the integral round y tends to zero as p-> 0. Thus r J o {Residue of z- l (l+z)~ l at z = -1}. 1+3? 1- At z = 1 we have r = 1, $ = w , and so, for the residue, lim f , % z - 1 ) , , . . g -*-l 1 (1+z) f = ( I)*"" 1 = e (a ~ 1)7r< = Hence f * &~ l , . f e 1 " 7 ' Jo 1+5* = ~ 27rt I !=* Bin air This integral can also be evaluated by integrating ) t using as contour a large semicircle in the upper half plane and the real axis indented by semicircles at 2 = and at z = 1. In this case the cut plane is unnecessary. The evaluation of the integral by this second method is left as an exercise for the reader.* By this second method one obtains the further result that f*&-*dx =77 Jo 1-* COtOTT. 48. Use of Contour Integration for deducing Integrals from Known Integrals The contours used so far have been either circles or semicircles, and although a large semicircle in the upper half plane is generally used for integrals of the type discussed in 45, there is no special merit in a semicircle. The rectangle with vertices <R, 7J+iJ? could also be * Bee Copaon, Functions of a Complex Variable, p. 140. THE CALCULUS OF RESIDUES 129 used in these cases. We now give two examples of deducing the values of some useful integrals by integrating a given function round a prescribed contour. We use, in the first case, a rectangle and in the second a quadrant of a circle. Example 1. Prove that f e~ x% cos 2ax dx . \ ^ire-** by J o integrating -* round the rectangle whose vertices are 0, R, ~ , ia. y C B Fio. 10. Let A be (1?, 0) and O be (0, a) in the Argand diagram. On OA, z = x ; on AB 9 z = R+iy ; on BG 9 z = x+ia ; and on 0(7, z = iy. Now e"** has no poles within or on this contour and so, by Cauchy's theorem, [ R f* f f 'o JQ JR Ja Hence 0. p /j e~ x *dx e a * Jo 'o Now 2oa? t sin t-ij a &'dy=*0. . (1 i f e-^a-zfly+if 1 ^ <e-^ $ . e a *. a Jo and so this integral -> as R > oo. On using the result that I e~ x *dx = i\/7r, we find, on making !?-> oo and equating real parts in (1), 1 cos r e~* Jo 130 FUNCTIONS OF A COMPLEX VARIABLE Example 2. By integrating e^'z - 1 round a quadrant of a circle of radius R, prove that, if 0<a< 1, _1 cos The contour required is drawn in fig y cos FIG. 17. Since the origin is a branch-point for the function z*- 1 , we enclose it in a quadrant of a small circle y of radius p. We integrate round the contour in the sense indicated by the arrows. On y, z = peiO and we get If <p a J dO = since | e~P sin e \ < 1 when p is small. It follows that /B->0, i/a>0. If a< 1, | z a ~ l \ < when J? is large enough, and so by the same argument as was used in proving Jordan's lemma ( 46) we have f J as #->oo, if a< L Hence, if 0<o< 1, we get, on making p-> and f J 0. THE CALCULUS OF RESIDUES 131 since there are no poles inside the contour. Hence r 00 r 00 / TTQ ira\ x*~ 1 (cos x+i sin x)dx = e~*y*~ l (cos +i sin I dy. JQ J \ 2 27 Since f tr*y*- l dy = r'(a),* on equating real and imaginary Jo parts, the required results follow. 49. Expansion of a Meromorphic Function Let f(z) be a function whose only singularities, except at infinity, are simple poles at the points z = a l9 z = a a , z = a 8 , ... ; and suppose that Suppose also that we know the residues at these poles : let them be b l9 6 2 , 6 3 , .... Consider a sequence of closed contours, either circles or squares, C lt (7 2 , <7 3 , ..., such that G n encloses a l9 a a , ..., a n but no other poles. The contours C n must be such that (i) the minimum distance R n of C n from the origin tends to infinity with n 9 (ii) the length L n of the contour C n is 0(R n ), (iii) on C n we must have /(z) = o(R n ). Condition (iii) would be satisfied if /(z) were bounded on the whole system of contours G n . When these conditions are satisfied we can prove that, for all values of z except the poles themselves, 1 1 To prove this, consider the integral wh^re z is a point within C n . The integrand has poles at the points a m with residues b m l{a m (a m z)} ; at = z * G.I., p. 84. 132 FUNCTIONS OF A COMPLEX VARIABLE with residue f(z)/z ; and at = with residue /(0)/z. In particular cases the last two residues may be zero. Hence If now we can prove that J->Q as w->oo, the theorem is proved. Here we require the conditions laid down above on the contours <7 n . On making use of these, we see that The series is uniformly convergent in any finite region which does not contain any of the poles. As an example of this theorem we prove that 1 oo / )n-l cosec z = 2z S -y-^ - . Consider the function /(z) = cosec z (z^0),/(0) = 0. Now z sin z has simple zeros at the points z=n7r,(n=... 2, 1, 1, 2,...) and so/(z) = : will have simple poles at those points. z sin z The residue at z = HIT becomes, on writing znir = f, sintf+nrr)} ^ i im (f +nir) ain (t+nn) f-^0 (f-f nn) cos (f +nir)+ am (f -j-nw) nir cos WTT There is no singularity at z = since z~sinz 0(1 z I 1 } zamz ~ z*+0{\z\ 4 } 0(\z\). Let O n be the square with corners at the points (n+ J)( 1 *)ir. The function 1/z is certainly bounded on these squares. To prove that cosec z is also bounded, consider separately the THE CALCULUS OF RESIDUES 133 regions (a) y>^ 9 (b) y<-ln, (c) -fr^y^n. In (a) we have y>i?r and coseo z for | 6 "-a-*| > |{| a"| - | a-* |}| - |{| a-f | - | c'\} |; and a similar argument applies to (b), writing y = t so that $> JTT. For (c), let AB be the line joining the points \ir,\Tri. Since | sin z \ = (cosh 2 !/ cos 2 #)* ; on AB we have, since x = JTT, | sin z | = cosh y^l, so that | cosec z \ ^1. Since cosec z has period w, it is bounded on all the lines joining (n + $i)n and (n + $ + $i)ir. Hence cosec z is bounded on all the squares (?. The previous theorem therefore gives cosec z- - I 1 \ + , ~n7T HTT/ the accent indicating that the term n = is omitted from the summation. Since the series with n>0 and with n<0 converge separately, we may add together the terms corre- sponding to n and write the expansion ft cosec z -- =* 2z z - n w~" 50. Summation ol Series by the Calculus ol Residues The method of contour integration can be used with advantage for summing series of the type 2f(n), if /(z) be a ineromorphic function of a fairly simple kind. Let be a closed contour including the points m, m+1, ..., ft, and suppose that /(z) has simj points a l9 a 2 , ..., a fc , with residues b v the integral / 77 COt7TZ/(z) o 134 FUNCTIONS OF A COMPLEX VARIABLE Tho function IT cot TTZ has simple poles inside C at the points z = m, ra+1, -., w, with residue unity at each pole. The residues at these poles of TT cot 772: /(z) are accordingly /(m), /(m+1), ...,/(n). Hence, by the residue theorem, f f(z)7TCotirz dz = 27Ti{f(m)+f(m+I)+...+f(n) J +6x77 cot 7ra 1 +... +6^71 cot TraJ. If conditions are satisfied which ensure that the contour integral tends to zero as w->oo, we can find the sum of the series Ef(ri). Suppose that /(z) is a rational function, none of whose zeros or poles are integers, such that z/(z)-> as | z |->oo. Let C be the square with corners (n+\)(\i). We have seen that cotTrz is bounded ... , I f fl . , dz 7rML on this square and so zf(z)ir cot TTZ \Jo z f for n R large enough, where M is the upper bound of | cot TTZ | on (7, L is the length of C and R is the least distance of the origin from the contour. Since L = 8R, the integral tends to zero as w->oo, and so n lim S f(m) = Trtyi cot 7ra 1 + . . . +6 fc cot 7ra k }. If we use 77 cosec TTZ instead of 77 cot m, we can obtain similarly the sums of series of the type Z( l) m f(m). oo I oo ( l) n Example. Find the sums of the series E -, E -. For the first, /(z) = and so z/(z)-> as | z |->oo. The z a +a a two poles of /(z) are at z = ai and the residues at these poles are l/2af. Hence 00 1 I ] I "\ _ 27 r ; SB w i r . cot Trai -; cot ( nai) \ = - coth we !.- m 2 +a 2 t^a* 2ai J a or -^ +2< THE CALCULUS OF RESIDUES 135 Similarly we get, by using IT cosec irz instead of w cot nz 9 oo / l\m I Z = T- . -4- r- cosech iro. m m2 + a 2o 2a In simple cases we can deal similarly with functions f(z) which have poles which are not simple. As an example, consider the series s i -.(+)' Here f(z) = (a+z)~ 2 has a double pole at z = a. By Taylor's theorem cot wz cot( 7rGt)4*(w2J+wct){ cosec f ( ira)}-f-..., and so the residue of cot irz/(z+a) 2 at = a is w cosec'Tra. oo l Hence 27 = w 2 cosec 1 wa. EXAMPLES V Use the method of contour integration to prove the following results 1 to 10 : , C* a dS L oos>30 . ( ,.v, ) - f (napositive +2 cos V5 integer). ir (6+2c) r ' J - cos a? 136 FUNCTIONS OF A COMPLEX VARIABLE cos ax dx ir . f Jo 73 f<* a^-icfo 2w /27ra + 7r\ J 1+5+? = 73 cos Hi-) coseo - ' (0<0<2) ' ++ 3 x a dx nil a) / rf z 11. Evaluate I - taken round the ellipse whose equation r c?2J is x % xy+y* +x+y = 0. Evaluate similarly - 1 round J 1+2 the ellipse 12. Show that the function }(z) = z/(ae" iM ) has simple poles at the points z = i loga+27rn, (n = 0, 1, 2, ...) ; and by integrating /(z) round a rectangle with corners at ir, .7r+in prove that, if a>l, fir xsinxdx n , 1+a I - - log - J 1+a 2 2a cos a; a a 13. By taking as contour a square whose corners are N, N+2Ni, where N is an integer, and making N-+CQ, prove that f 00 dx 10 I - = log 2. Jo (l+# 2 ) cosh (jTTtf) 6 14. By integrating e-*^* 1 " 1 round a sector, of radius R 9 bounded by the lines arg z = 0, arg z = a< JTT (indented at 0), prove that, if fc>0, n> 0, r cos cos J - r> x*dx 7T\/3 15. Prove that P -3 r = r- J o a^ l o 16. Prove that ____&- _(J-a+l-ar-), (o>0). *a*+x* 2o* THE CALCULUS OF RESIDUES 137 17. By integrating e a< /(e~ 2<1 1) round a suitable contour, prove that I : L. = ITJ. coth fact J e*y-l * * 2a 18. Prove that sees = 47727(--l) n (2n + l)/{(2n-fl)*77 a --42*}. o 19. Prove that, if 77<o<77, sin 02 2 *J n sin na ~: = - 27 ( l) n - , sin 772 77 n i 2' n* cos 02 1 22 cos no __ == -| 2i ( 1) sm 772 772 77 na ,i z*n* 20. Prove that * 1 __ 77 sinh(77aV2)+sin (77a\/2) n--ao nT+a* a 3 V 2 cosh (770 V 2 )~ oo / !\n and find 27 ~ ^ . (ii) Prove that 00 1 1 77 27 - - = - i ; (coth 77a+cot na). nl w o 2a 4 4a 3 77 sin az , . _ _ 21. By integrating - - round a suitable contour, 2 3 Sin 772 prove that 1_I + I_I+ ^l 8 P 3 3 5 8 7 3 32' f 2+1 22. (i) Prove that 2 2 log - d2, taken round the circle J 2 1 2 | = 2, has the value 477^/3. f lg x dx 77 (u) Prove that ~ - ~- = -- , using as contour a J (l+ar) a 4 arge semicircle in the upper half-plane indented at the >rigin. MISCELLANEOUS EXAMPLES 1. If u> = tt+tt> is a regular function of z, show that / d* d 2 \ I + 1 w = is equivalent to dzcz Hence show that iv\'(z+z) 9 !.(-)} #)-#*) +c f \& 41 J where C is an arbitrary constant.* Use the above relations to find the regular function / (z) for which u is (i) log (x* +t/ 2 ), (ii) x 2 y 2 +4txy and for which v is e 2x (y cos 2y+x sin 2y). 2. If x = r cos 0, y = r sin change the independent / a 2 d* \ variables in I + 1 <f> = 0, to r, 8. If u (x 2 +y 2 )*/(x*y*) find the function <f>(u) which satisfies v a < = 0. Find also the regular function / (z) of which <f>(u) is the real part. 3. Show that, if y +0, there are two points unaltered by the transformation unless (8 a) 2 +4y = 0, in which case there is only one such point. Show that, if z = I is this point, -L L+ ,_! s-1 +" where K is a constant. * This result was communicated to me by Prof. A. Oppenheim- 138 MISCELLANEOUS EXAMPLES 139 Show that z = 1 is the only fixed point of the trans- formation l+iz tan A w l+i tan A* (0<A<j7r), and that this transformation maps the inside of | z | = 1 on the inside of | w \ = 1. Sketch the curve in the w-plane which corresponds to the straight line joining 2 = 1 and z = t. 4. Prove that, if fc>0, w = tan (irz/lk) maps lines parallel to the axes in the z-plane on systems of coaxal circles in the u?-plane. Find what corresponds to the infinite strip t = & and indicate in a figure the region of the w-plane corresponding to the square Q^x^k, k^y^2k. 5. Show that, if a>0, the relation u> = aicot \z maps the semi-infinite strip 0^a5^2?r, t/^0 on a half -plane cut from u = a to w = oo . Two circles, with real limiting points (a, 0), are drawn in the cut w-plane with centres (2ka, 0), (to, 0), where Jfe>l. Show that the space between these circles is mapped on the interior of a rectangle in the s-plane whose area is (2k-l)(k + I) " g 6. Express the transformation in the form - = k\ - - 1 and hence show that the inside w ft \z b! of | z | 1 is mapped on the whole w-plane cut along a segment of the real axis. Illustrate by a diagram what corresponds in the u;-plane to that part of the circle | z i \ = <\/2 which lies in the fourth quadrant. 7. If a, 6, c, d are real constants, some of which may be zero, and az*+bz+o *= z+d ' 140 FUNCTIONS OF A COMPLEX VARIABLE show that there are two values of w, each of which corresponds to a pair of equal values of z ; and that these values of w can only be equal if the transformation is bilinear. Discuss the transformation -2s w-- 1-23 in this way, and find the boundaries in the z-plane which correspond to| w \ = 1. Show in a diagram the regions of the z-plane corresponding to |10|<1. 8. Iff (z) is regular within and on the circle | z \ = R and ^ If ( z ) I <M on ^ e circle, prove that, if | z \ = r<R 9 oo where a is the constant term in / (z) = Z a^s*. =*o If R = 1, a = 1 and \f (z) \<k on \z\ = I, prove that / (z) does not vanish within the circle | z \ = 1/(1 +&). 9. Show that, if 0<a<7r and 4az cot a then w f (z) gives a conformal transformation when z lies in any finite region excluding the points z = t, z = cot a, z = tan a. Show that the boundary of the semicircle | z \ = 1, Rz>0 corresponds to an arc of a circle in the w-plane subtending an angle 4a at the centre. Find the two points in the z-plane corresponding to the centre of this circle. 10. Show that, if u? = two finite points of the z-plane are mapped on every finite point of the to-plane, except the origin and w = 1, and explain why the mapping ceases to be conformal at the point z = 1. Show, in a diagram, the two domains of the z-plane which are mapped on the semi -circular domain | w |<1, Iw>0. MISCELLANEOUS EXAMPLES 141 oo oo 11. Iff(z) = 2a n z n and <f>(z) = b n z n are both regular within a domain D which includes the circle | z \ = 1, prove that o * o Hence, or otherwise, prove that <W- " (IT) 5 + dV- " " s; J7 COS (2 8in e)de - 12. A function <(z) is regular over the whole z-plane, except at z = and at z = oo, and, for all values of z 9 <f>(z) z<l>(pz), where |)5|<1. Find the Laurent expansion of (f>(z), given that the constant term in the expansion is unity. Show that <() =0. 13. ABCD is a square whose vertices are 0, t, 1 +i, 1 ; F t is the line BG and F 2 consists of the other three sides of the square. Iff (z) = 2 5 +z 2 + 1 prove that (i)/ (z) does not assume any real non-negative value on JTj (ii) R/ (z) >0 at all points of r a except B. Hence, or otherwise, evaluate A and deduce that /(z) has just one zero inside the square ABGD. 14. By integrating e ij>f /cosh a z along the lines I(z) = 0, I(z) = TT prove that oo COS px irp cosh 2 x 2 sinh \j 15. By integrating cosec z round a square whose vertices are (n + J)(7r^) prove that cosec t 2et * ( l) n " /i_iw 142 FUNCTIONS OF A COMPLEX VARIABLE 16. By discussing dz L (-7r<a<7r) cos TTZ z round the circle | z \ = r, where r is an integer, show that cos q = 2 (-l)f+l(r-}) CO s(f--})a ~~ 17. By integrating z~* sec \itz sech \TTZ round the square Rz = 2N, Iz 2N, where N is a positive integer, and making N-+CQ prove that sech sech - sech 5 16 35 I 55 18. By considering coth TTZ cot TTZ , taken round the square a;= (N+$), y == (^+|) where W is a large positive integer, prove that ^r 77T 8 n INDEX The numbers refer to the pages Aerofoil, 76 Algebra, the fundamental theorem of, 113 Analytic continuation, 103-107 Analytic function, 105 Argand diagram, 6 Argument, 6; principle of the, 108 Bilinear transformation. (See Mobius) Biuniform mapping, 33 ; by Mobius* transformations, 46 Boundary point, 8 Bounded set of points, 8 Branch of function, 26 Branch -point, 28 Calculus of residues, 114-137 Cardioid, 62, 77, 81 Cauchy's theorem, 89-92 ; integral, 93 ; residue theorem, 116 Cauchy-Riemann equations, 12, 30, 36, 89 Circle of convergence, 1 7 Circular crescent, 53, 68 ; sector, transformations of, 69 Coaxal circles, 46-48 Complex integration (see Con- tour integral), 85 Complex numbers, defined, 1 ; modulus of, 2 ; argument of, 6 ; real and imaginary parts of, 1 ; geometrical representation, 6 ; ab- breviated notation for, 3 Confocal conies, 71, 74, 81, 82 Conformal transformations, definition of, 32 ; tables of special, 78, 79 Conjugate, complex numbers, 4 ; functions, 14 Connex, 9 143 Continuity, 9 Contour, 86 ; integral, definition of, 88; integration, 117, 128 etseq. Critical points ol trans- formations, 37, 45, 65 Cross-cut, 93 Cross-ratio, 49 Curve, Jordan, 8 ; Jordan curve theorem, 9 Cut plane, 27, 69, 70, 73, 126 Definite integrals, evaluation of, 114-131 Determinant of transformation, 41 Differentiability, 10-11 Domain, 9 Element of an analytic function, 105 Elementary functions, 17-25 Ellipse, 72, 74 Equation, roots of, 1, 31, 109, 113 Equiangular spiral, 31, 82 Expansion, of function in a power series, 18, 95 ; of meromorphic functions, 131 ; Laurent's, 97 Exponential function, 20, 31 Function, analytic, 11, 105; hyperbolic, 23 ; holo- morphic, 11; integral, 21; logarithmic, 24 ; many- valued, 26 ; meromorphic, 107, 131 ; rational, 20, 103; regular, 11 Gamma function, 130 Goursat's lemma, 91 Green'* theorem, 89 144 INDEX Harmonic functions, 37, 111 Hyperbolic functions, 23 Imaginary part, 1 Infinite integrals, evaluation of, 119-130; principal value of, 120 Infinity, point at, 9, 102 Integral functions, 21 Interior point, 8 Invariance of cross-ratio, 49 Inversion, 43 ; transformation, 55 Isogonal transformation, 32 Jordan curve, 8 Jordan's lemma, 122 Laplace's equation, 15, 30, 54 Laurent expansion, theorem, 97, 112 Lima9on, 61 Limit point, 8 Liouville's theorem, 96 Logarithm, 21 Logarithmic function, 24 Magnification, 34, 39, 41 Many-valued functions, 26 Mapping, 32 Maximum modulus, principle of, 109 Meromorphic function, 107, 131 Milne -Thomson's construction, 16 M6bius' transformations, 40-56 ; some special, 51-54 Modulus, 2 de Moivre's theorem, 6 Neighbourhood, 7 One-one correspondence, 33 Parabola, 62, 64 Point at infinity, 9, 102 Poisson's formula, 111 Pole, 99, 107 Power series, 17-25 Principal part at pole, 99 Principal value, of arg z, 6 ; of log z, 25 ; of infinite integral, 122 ; of f*, 25 Radius of convergence of power series, 17 Rational functions, 20, 103 Real part, 1 Rectangular hyperbolas, 61 Rectifiable curve, 86 Regular function, 1 1 Residue, 99; theorem, 114; at infinity, 114 Riemann surface, 27, 59-60; 63, 71, 73, 75 Roots of equations, 1,31,109, 113 Rouche"'s theorem, 108 Schwarz-Christoffel transforma- tion, 80 Sets of points, in Argand diagram, 7-9 ; bounded, 8 ; closed and open, 8 Singularity, 11, 99; essential, 99; isolated, 99; non- isolated, 101 of analytic function, 105 Spiral, equiangular, 31, 82 Successive transformations, 58, 83 Summation of series, 133 et seq. Taylor's theorem, 19, 96-96; series, 101, 112 Term-by-term differentiation, 19 Transformations, conformal, 32 et seq. ; isogonal, 32 ; special, 57-83 ; tables of, 78, 79 Trigonometric functions, 23 Uniform continuity, 10, 87 Upper bound for contour integral, 88 Vectorial representation of com- plex numbers, 6 Weierstrass' definition of ana- lytic function, 105 Zero, 98, 107